Jan cka Classes of rings determined by a categorical property The work [71] is devoted to study of...

HABILITATION THESIS

Jan �Zemli�cka

Classes of rings determinedby a categorical property

Department of Algebra

Prague 2016

To Lenka, Ivan, Nina, Antonie, and Eli�ska.

Contents

Preface 2

1 Introduction 3

2 Self-small modules and strongly steady rings 26

3 Small modules and steady rings 43

4 The defect functor of homomorphisms and direct unions 62

5 Re ection of categorical properties to a ring structure 93

1

Preface

The core of the presented habilitation thesis consists of the following articles:

A. Simion Breaz, Jan �Zemli�cka, When every self-small module is �nitely gen-

erated, J. Algebra 315/2 (2007), 885{893.

B. Jan �Zemli�cka, When products of self-small modules are self-small, Commun.Algebra 36/7 (2008), 2570{2576.

C. Jan �Zemli�cka, Steadiness is tested by a single module, in: Kelarev, A. V.(ed.) et al., Abelian groups, rings and modules. Proceedings of the AGRAM2000 conference, Perth, Australia, July 9-15, 2000. American MathematicalSociety (AMS), Providence, RI. (2001), Contemp. Math. 273, 301{308.

D. Jan �Zemli�cka, Steadiness of regular semiartinian rings with primitive factors

artinian, J. Algebra 304/1 (2006), 500{509.

E. Simion Breaz, Jan �Zemli�cka, The defect functor of homomorphisms and

direct unions, Algebr. Represent. Theor. 19/1 (2016), 181{208.

F. Jan �Zemli�cka, Socle chains of abelian regular semiartinian rings, J. PureAppl. Algebra 217/6 (2013), 1018{1025.

G. Tom�a�s Penk, Jan �Zemli�cka, Commutative tall rings, J. Algebra Appl., 13/4(2014).

H. M. Tamer Kosan, Jan �Zemli�cka, Mod-retractable rings, Commun. Algebra42/3 (2014) 998-1010.

I. M. Tamer Kosan, Jan �Zemli�cka, On modules and rings with restricted min-imum condition, Colloq. Math., 140,1 (2015), 75{86.

2

Chapter 1

Introduction

This chapter contains a survey summarizing several particular concepts and toolsuseful in research of connections between a ring structure and a structure ofcategories of modules.

3

CLASSES OF RINGS DETERMINED BY A CATEGORICAL

PROPERTY

There are many properties of the category of all modules over a ring which can be

easily recognized from the structure of rings. This phenomenon can be illustrated

by the classical theorem characterizing left perfect rings:

Theorem 0.1. [17, Theorem P] The following statements are equivalent for a ring

R:

(1) Every left module has a projective cover.

(2) R=J(R) is semisimple and J(R) is left T-nilpotent .

(3) R=J(R) is semisimple and every non-zero left R module contains a maximal

submodule.

(4) R satis�es the descending chain condition on principal right ideals.

(5) Every at left module is projective.

Note that the conditions (1) and (5) deals with the structure of the category of

all modules over the ring, however the conditions (2), (3), and (4) are expressed in

the language of ring structure.

Characterization by both ring-theoretical and categorical properties are known

for various classical classes of rings such semisimple, hereditary, semihereditary or

abelian regular ones. Recall for example a characterization of von Neumann regular

rings [49, Theorem 1.1 and Corollary 1.13] which appears to be useful as a test class

for ring theoretical characterization of some categorical properties.

Theorem 0.2. The following statements are equivalent for a ring R:

(1) For every x 2 R there exists an element y 2 R such that x = xyx.

(2) Every principal left (right) ideal is generated by an idempotent.

(3) Every �nitely generated left (right) ideal is generated by an idempotent.

(4) Every �nitely generated submodule of a projective left R-module P is a direct

summand of P.

(5) Every left (right) module is at.

Clearly, the �rst three condition are ring-theoretical and the last two module-

theoretical. Furthermore, it is worth mentioning that the �rst condition quanti�es

only elements of a ring while the second and third ones are formulated in language

of lattices of one-sided ideals.

Unfortunately, not all classes of rings de�ned by some natural condition on cate-

gory of modules can be described by some nice ring-theoretical property. There are

two reasons of such a lack. The �rst one is caused by our ignorance; the problem

seems to be simply too hard for our imperfect tools and the goal of this thesis is to4

CLASSES OF RINGS DETERMINED BY A CATEGORICAL PROPERTY 5

at least partially solve some problems of the described character. The second rea-

son is fundamental, based on set theory. Such an example of a module-theoretical

properties which cannot be describe in the ring structure example is the existence

of Whitehead test module for projectivity. As it is proved in [96] it is independent

of ZFC with Generalized Continuum Hypothesis over all right hereditary non-right

perfect ring.

The main objective of this thesis is to partially summarize known results for

several classes of rings determined by some categorical property. Four of the studied

classes, namely of steady, strongly steady, tall, and mod-retractable rings represent

typical examples of such rings, since their categorical de�nitions are natural an

easily applicable, while other two classes of general semiartinian, and RM-rings can

be de�ned by ring-theoretical property, nevertheless relevant structural questions

coming from the context of module theory needs transfer categorical properties to

the ring structure.

Let us remark that a module means right R-module over some unitary associative

ring R within the whole text. For non-explained terminology we refer to standard

monographs [11, 49, 90].

1. Compact objects

An object c of an abelian category closed under coproducts and products is said

to be compact if the covariant functor Hom(c;�) commutes with all direct sums i.e.

there is a canonical isomorphism in the category of abelian groups Hom(c;LD) �=L

Hom(c;D) for every system of objects D. The concept of compactness presents

an easy way to replace �nitely generated modules in general abelian categories.

Nevertheless, the clear form of the categorical de�nition is a reason why compact

objects can be applied as a useful tool also in categories containing �nitely generated

objects in standard sense.

1.1. History. The systematic research of compact objects in the context of module

categories was started by Hyman Bass in 60's. His famous book [18] contains as

an exercise a basic non-categorical characterization of the notion. Let us mention

here the author's comment to the exercise that examples of compact objects in the

category of all modules which are not �nitely generated "are not easy to �nd" [18,

p.54].

The introductory work on theory of compact modules is due to Rudolf Rentschler.

However his PhD thesis [78] and the paper [79] contains a list of basic examples

and several necessary and su�cient conditions of compactness, the core of these

works is an attempt to answer the natural question over which rings coincide the

classes of compact modules and of �nitely generated ones. It should be mentioned

that compact objects in categories of all modules have been studied under various

terms: module of type �, dually slender, �-compact, or U-compact module. We

will use the term small module here.

6 CLASSES OF RINGS DETERMINED BY A CATEGORICAL PROPERTY

Further study of small modules has been motivated by progress of research in

several di�erent branches of algebra. One of the most important source of ques-

tions concerning smallness or more generally compactness comes from the context

of representable equivalences of module categories. However ?-modules which ap-

pear to be an important notions in this branch of module theory were shown to

be necessary �nitely generated [32, 33], more general context deals with in�nitely

generated small modules [93, 94, 95]. Another important motivation for study of

the notion has appeared in the structure theory of graded rings [75] and almost free

modules [94]. Lattice theoretical approach to smallness is presented in the work

[54]. Fruitful motivation of many questions in this theory comes from the dual

context of so called slim and slender modules [38, 42, 45, 74], albeit the tools and

ideas of research are far from being dual.

Commuting properties of functors Hom are studied in many cases only for mod-

ules from the category Add(M) of direct summands of direct sums of copies of

some module M . Recall that a module M which is a compact object of the cat-

egory Add(M) is called self-small. The notion was introduced in [12] as a tool

for generalization of Baer's lemma [48, 86.5]. Self-small modules turn out to be

important also in the study of splitting properties, [5, 21] and representable equiva-

lences between subcategories of module categories in connection with tilting theory

[32, 33]. The notion is very useful in structure theory of mixed abelian groups

[9, 20].

The work [71] is devoted to study of compactness in stable categories, i.e. cat-

egories whose objects are all modules and groups of morphisms factorize through

projective modules. Namely, it is proved that over right perfect rings compact of ob-

jects of the stable category can be represented by some standard �nitely generated

modules.

1.2. Small modules. As it is shown in [18] or in [79, 1o], small modules can be

described in natural way by language of systems of submodules.

Lemma 1.1. The following conditions are equivalent for an arbitrary module M:

(1) M is small,

(2) if M =Si<!Mn for an increasing chain of submodules Mn �Mn+1 �M ,

then there exists n such that M =Mn,

(3) if M =P

i<!Mn for a system of submodules Mn � M , n < !, then there

exists n such that M =P

i<!Mn.

The condition (2) implies immediately that every �nitely generated module is

small. Moreover, it is clear from (3) that there is no in�nitely countably generated

small module. An another easy consequence of Lemma 1.1 is an observation that

a union of strictly increasing chain of the length �, for an arbitrary cardinal �

of uncountable co�nality, consisting of small submodules give us as well a small


module. A small module can be constructed in such a way for instance as a �-

generated uniserial module, which is, indeed, the union of a chain of �-many cyclic

submodules.

This construction motivates de�nition of particular subclasses of small modules.

For an arbitrary cardinal number � we say that a module M is �-reducing if for

every submodule N � M such that gen(N) � � there exists a �nitely generated

submodule F such that the inclusions N � F �M holds.

For an arbitrary ring R let us denote by SM(R), R�(R), FG(R) and FP(R)

respectively the classes of all small, �-reducing, �nitely generated and �nitely pre-

sented right R-modules. It is easy to formulate the following hierarchy of these

classes:

FP(R) � FG(R) � R�(R) � R�(R) � SM(R)

where � < � are in�nite cardinals.

Note that all the inclusions are strict in general. Of course, every �+-generated

ideal in valuation ring is a witness of the inequalities R�+(R) 6= R�(R) 6= FG(R).

Furthermore, it is proved in the paper [95, Theorem 2.8] that a ring power F! for

each �eld F contains a small right ideal which is not !-reducing. It is important to

remark that classes R�(R) and SM(R) have similar class properties as the class

FG(R). Namely, the classes SM(R) and R�(R) for each in�nite cardinal are closed

under taking homomorphic images, extensions and �nite sums [104, Proposition

1.3].

There exist natural classes of rings over which each injective module is necessary

small. Let us denote by I(R) the class of all injective modules over a ring R.

Theorem 1.2. [95, Theorem 1.6] Let � be an in�nite cardinal and R a ring.

(1) If there exists an embedding R(�)R ! RR, then I(R) � R�(R).

(2) If there exists an embedding R2R ! RR, then I(R) � R!(R).

The hypothesis of (1) is satis�ed by the endomorphism ring End(V ) for any

�-dimensional vector space V . Furthermore, any non-commutative domain which

does not satisfy the right Ore condition (for example polynomials in two non-

commuting variables Zhx; yi) satis�es the hypothesis of (2).

A similar observation as in Theorem 1.2 is made in the paper [33]:

Proposition 1.3. [33, Lemma 1.10] Let R be a simple ring containing an in�nite

orthogonal set of idempotents. Then I(R) � R!(R).

As a consequence we get that I(R) � R!(R) for every non-artinian simple von

Neumann regular ring R. Indeed, it means that if all injective modules are small,

then there exists a proper class of non-isomorphic small modules. Thus we have

examples of rings over which small modules can be arbitrarily large.

1.3. Steady rings. Rings over which the class of all compact (or small) modules

coincides with the class of all �nitely generated ones are called right steady. It is


well-known that the class of all right steady rings is closed under factorization [33,

Lemma 1.9], �nite products [94, Theorem 2.5], and Morita equivalence [43, Lemma

1.7].

Clearly, rings over which there exists proper class of non-isomorphic small mod-

ules (as those from Theorem 1.2 and Proposition 1.3) are not steady. On the other

hand, several classes of rings satisfying some �niteness conditions are well-known

to be right steady:

Theorem 1.4. A ring R is right steady provided any of the following conditions

holds true:

(1) R is right noetherian; ,

(2) R is right perfect;

(3) R is right semiartinian of �nite socle length,

(4) R is a countable commutative ring,

(5) R is an abelian regular ring with countably generated ideals.

Note that (1) has been established independently by several authors ([79, 70],

[32, Proposition 1.9], [44, p.79], (2) is proved in [33, Corollary 1.6], (3) in [95,

Theorem 1.5], (4) in [79, 110], and (5) in [113, Corollary 7].

As an easy consequence of the Theorem 1.4(1) we obtain a characterization of

rings over which small modules are precisely �nitely presented ones:

Theorem 1.5. [104, Theorem 1.4] A ring R is right noetherian if and only if

SM(R) = FP(R).

Although an existence of a general ring-theoretic criterion for steady rings is still

an open problem, there is a construction of some kind of minimal example of an

in�nitely generated small module over a non-steady ring:

Theorem 1.6. [102, Theorem 1.4] Let R be a ring, � = card(R)+, and denote by

Simp the representative set of all simple right modules. Then R is not right steady

if and only if T =QS2Simp S

� �L

S2SimpE(S) contains an in�nitely generated

small submodule.

Obviously Theorem1.6 can be reformulated to the claim that a ring R is right

steady if and only if the module T (of cardinality bounded by 22card(R)

) contains

no in�nitely generated small submodule.

Let us remark that for commutative regular rings a module-theoretical criterion

of existence of an in�nitely generated small module can be formulated in a more

elegant form, that the representative class of small modules over a commutative

regular ring is in general a set, and there is an estimate of the cardinality of each

small module:

Theorem 1.7. [102, Theorem 2.7] Let R be a commutative regular ring. Then

R is steady if and only if the module R� = HomZ(R;Q=Z) contains no in�nitely

generated small submodule.


We have remarked that a ring-theoretical characterization of steadiness is an

open problem, nevertheless criteria of steadiness are known for several particular

classes of rings.

Remind that a ring is called right semiartinian if every non-zero cyclic module

contains a simple submodule [72]. Large classes of examples both steady and non-

steady abelian regular semiartinian rings are constructed in the paper [43]. The

articles [83] and [105] characterize steadiness of abelian regular semiartinian rings

[83, Theorem 3.4] and regular semiartinian rings with primitive factors artinian:

Theorem 1.8. [105, Theorem 3.5] Let R be a regular semiartinian ring with prim-

itive factors artinian. Then the following conditions are equivalent:

(1) R is right steady;

(2) R is left steady;

(3) There exists no in�nitely generated small right ideal of any factor of R.

(4) There exists no in�nitely generated small left ideal of any factor of R.

In particular, an abelian regular semiartinian ring R is is not right steady if and

only if there is an abelian regular factor-ring, �R, of R and a member, I, of the socle

chain of �R such that I is an in�nitely generated dually slender right �R-module [83,

Criterion A].

In the case of abelian regular rings the criterion of steadiness is formulated in

the work [109] where w(M) = supfdimR=I(M=MI)j I maximal idealg:

Theorem 1.9. [109, Theorem 3.2] Let R be an abelian regular ring. Then the

following conditions are equivalent:

(1) R is right steady,

(2) R=Tn<! In is right steady for every system of maximal ideals In, and there

exists no small module M with �nite w(M) which is either !1-generated or

contained inQi<! Fi where Fi are n-generated modules.

(3) There exists no (!1-generated) !1reducing module and no in�nitely gener-

ated small submodule ofQn<! R=Jn for any system of ideals Jn.

(4) There exists no in�nitely generated small submodule ofQn<! R=Jn for any

system of ideals Jn and every !1-generated module M with �nite w(M)

contains a countable set C such that M=Tc2C MAnn(c) is in�nitely gen-

erated.

The fact that steady continuous regular rings are precisely semisimple rings is

presented in [107, Theorem 4.7]. Furthermore, a necessary and su�cient condition

of steadiness of valuation rings is given in [113, Theorem 13] and more general case

of chain rings (i.e. rings with linearly ordered lattices of both right and left ideals)

is characterized in the paper [103]:

Theorem 1.10. [103, Theorem 2.4] For a chain ring R the following conditions

are equivalent:


(1) R is right steady.

(2) There exists no !1-generated uniserial right module.

(3) R=rad(R) contains no uncountable strictly decreasing chain of ideals, R

contains no uncountably generated right ideal and for every ideal I and for

every prime ideal P � I there exists an ideal K such that P � K � I.

However countable valuation rings are steady, it is known an example of a count-

able chain ring which is not right steady [103, Example 1.9]. The result of Theo-

rem 1.10 can be generalized for a class of serial rings:

Theorem 1.11. [103, Theorem 3.5] The following conditions are equivalent for a

serial ring R with a complete set of orthogonal idempotents fei; i � ng:

(1) R is right steady,

(2) eiRei is right steady for every i � n,

(3) there exists no !1-generated uniserial right R-module.

It is an open problem whether some analogue of Hilbert basis theorem is valid

for steadiness, i.e. whether a polynomial ring over a right steady ring is necessary

right steady. It is known for example that polynomial rings in �nitely many vari-

ables over right perfect ring [108, Proposition 2.6] and polynomial rings in count-

ably many variables over commutative neotherian rings are right steady [79, 11o],

but the question whether polynomial rings in countably many variables over non-

commutative neotherian rings are right steady waits for an answer. The strongest

result concerning countably many variables is the following claim:

Theorem 1.12. [108, Theorem 2.7] If X is a countable set of variables and R a

right perfect ring such that EndR(S) is �nitely generated as a right module over its

center for every simple module S, then R[X] is right steady.

On the other hand, polynomial rings in uncountably many variables are not

steady as it is witnessed by the following example.

Example 1.13. [108, Example 3.1, Proposition 3.2] Let R be an arbitrary ring

and consider the additive monoid N!1 . For every � < � � !1 de�ne e�� 2 N!1

by the rule e��( ) = 1 whenever 2 h�; �) and e��( ) = 0 elsewhere. Moreover,

E denotes the submonoid of N!1 generated by fe�� j � < � � !1g and consider a

monoid ring S = R[E]. Then the idealS�<! e0�S is !1-generated and !1-reducing

as a right S-module, which proves that S is not right steady.

LetX be an uncountable set of variables. Since there exists a surjective map ofX

onto the monoid E, it can be extended to a surjective homomorphism from the free

commutative monoid in free generators X to the monoid E and this homomorphism

of monoids can be extended to a surjective homomorphism of the polynomial ring

R[X] onto R[E]. Thus R[X] is not right steady.


1.4. Self small modules. Recall that a module M is self-small provided it is

a compact object in the category Add(M). Similarly as in the case of non-small

modules, non-self-small ones can be characterized by the condition that there exists

a countable chain M0 ( M1 ( � � � ( Mn ( : : : ; n < ! of submodules of M such

that M =Sn<!Mn and for every n < ! there exists a non-zero endomorphism

fn : M ! M such that fn(Mn) = 0 [12, Proposition 1.1]. It is worth mentioning

that the full endomorphism ring can serve as a tool for recognizing whether a

module is self-small. In particular, if for a module M either End(M) is countable

or the �nite topology on End(M) is discrete, then M is self-small [12, Corollaries

1.4 and 2.1]. Nevertheless, endomorphism rings cannot detect self-smallness of a

module in general:

Theorem 1.14. [106, Theorem 2.9] Let R be a non-artinian abelian regular ring.

Then there exists a pair of a self-small module M and a non-self-small module N

such that EndR(M) �= EndR(N).

The class of all self-small modules is closed under endomorphic images and direct

summands but the following example shows that it is not closed under �nite direct

sums:

Example 1.15. [40, Example 4] The groupQp2P Zp is self-small by [106, Example

2.7] as well as the group Q. The product Q �Qp2P Zp is not self-small by [40,

Example 3].

Let us remark that the natural question which �nite sums of self-small modules

are as well self-small has an easy answer. Note that the hypothesis on Hom-groups

in the condition (2) is satis�ed if for example Hom(Mi;Mj) = 0 whenever i 6= j.

Proposition 1.16. [40, Proposition 2.4] The following conditions are equivalent

for a �nite system of self-small modules (Mij i � k):

(1)L

i�kMi is not self-small

(2) there exist i; j � k and a chain N1 � N2 � ::: � Nn � ::: of proper

submodules of Mi such thatS1n=1Nn = Mi and HomR(Mi=Nn;Mj) 6= 0

for each n 2 N.

The case of in�nite products of self-small modules is much more complicated and

only particular results are known.

Proposition 1.17. [106, Proposition 1.6] Let (Mij i 2 I) be a system of self-small

modules satisfying the condition HomR(Qj2InfigMj ;Mi) = 0 for each i 2 I. ThenQ

j2I Mj is a self-small module.

It is well-known that over semisimple rings as well as over local or commutative

perfect rings the classes of small, self-small and �nitely generated modules coincides.

On the other hand, every generic module is an example of an in�nitely generated

self-small module over (of course artinian) Kronecker algebras.


It motivates the de�nition of right strongly steady rings as rings over which every

right self-small module is �nitely generated. Note that the ring R =

�Q R0 R

�is

non-singular right artinian but it is not right strongly steady since its maximal right

ring of quotients

�R RR R

�serves as an example of an in�nitely generated self-small

R-module [22, Example 3.11]. On the other hand, every upper triangular matrix

ring over a division ring (and, in particular, over a �eld) is right strongly steady

[22, Example 3.13].

Closure properties of strongly steady rings are similar as in the case of steady

ones; they include factorization, �nite products and Morita equivalence [22, Lem-

mas 2.1-4]. However the commutativity simpli�es the situation, ring theoretical

characterization of strongly steady rings is an open problem even in this case.

More clear is the (important) case of right non-singular rings:

Theorem 1.18. [22, Theorem 3.9] Let R be a right non-singular right strongly

steady ring. Then R is right artinian.

This result allows to formulate a criterion for commutative non-singular rings:

Theorem 1.19. [22, Theorem 3.10] A non-singular commutative ring is strongly

steady if and only if it is semi-simple.

Furthermore note that every right noetherian right strongly steady ring is right

artinian by [22, Proposition 3.16].

Special attention is given to study of self-small abelian groups. It is easy to see

that every self-small torsion group is �nite [12, Proposition 3.1], but the question

which mixed abelian groups are self-small seems to be very interesting and attractive

for researchers [5, 7, 20, 21]. If A is a torsion free abelian group of �nite rank, then

the R-typ of A is the quasi-isomorphism class of A=F , where F is a free subgroup of

A with A=F torsion. To conclude this section recall at least one basic result about

self-smallness of mixed abelian group with �nite rank torsion-free part:

Proposition 1.20. [12, Proposition 3.6] Suppose that A is a mixed abelian group

and that A=tA has �nite rank. Then A is self-small if and only if

(a) for all primes, p, (tA)p is �nite and

(b) the R-type of A=tA is p-divisible for all primes p with (tA)p 6= 0.

1.5. Abelian categories. However the de�nition of a compact object is categori-

cal, we have discussed results in the category of modules which can be formulated

just in the language of modules. Nevertheless, some particular questions of the

theory can be easily formulated in language of abelian categories. Before we try to

do it, let us start with needed categorical terminology and basic tools.

A category with a zero object is called additive if for every �nite system of

objects there exist product and coproduct which are canonically isomorphic, every

Hom-set has the structure of an abelian group and the composition of morphisms


is bilinear. An additive category is abelian if there exists kernel and a cokernel

for each morphism, monomorphisms are exactly kernels of some morphisms and

epimorphisms cokernels. A category is said to be complete (cocomplete) whenever

it has all limits (colimits) of small diagrams exist.

We suppose in the sequel that A is an abelian category closed under arbitrary

coproducts and products. By the term family or system we mean any discrete

diagram, which can be formally described as a mapping from a set of indexes to a set

of objects. Suppose thatN �M are two families of objects of the categoryA. Then

a corresponding coproducts are denoted by (LM; (�M jM 2M)), (

LN ; (e�N jN 2

N )) and a products by (QM; (�M j M 2 M)), (

QN ; (e�N j N 2 N )). Note that

there exists canonical morphisms �N :LN !

LM and �N :

QM!

QN given

by universal properties of colimitLN and limit

QN , which satis�es �N = �N e�N

and �N = e�N�N for each N 2 N .

For arbitrary ' = ('N j N 2 N ) 2LfA(M;N) j N 2 Ng let us denote by F

a �nite subsystem such that 'N = 0 whenever N =2 F and let � : M !QN be

the morphism given by the universal property of the product (QN ; (�N ); N 2 F)

applied on the cone (M; ('N j N 2 N )) (i.e. �N � � = 'N ). Then

N (') = �F � ��1 � �F � �

where � :LF !

QF denotes the canonical isomorphism. Note that the de�ni-

tion N (') does not depend on choice of F . Furthermore the mapping N is a

monomorphism in the category of abelian groups for every family of objects N .

Now, we are ready to formulate precise general de�nition of the central notion.

An object M is said to be C-compact if N is an isomorphism for every family

N � C. Note that the class of all C-compact objects is closed under �nite coproducts

and cokernels since the contravariant functor A(�;LN ) commutes with �nite

coproducts and it is left exact.

Now we are able to formulate an elementary criterion of compact object, which

generalizes Lemma 1.1:

Lemma 1.21. If M is an object and a class of objects C, then it is equivalent:

(1) M is C-compact,

(2) for every N � C and every f 2 A(M;LN ) there exists �nite subsystem

F � N and a morphism f 0 2 A(M;LF) such that f = �F � f 0,

(3) for every N � C and every f 2 A(M;LN ) there exists �nite subsystem

such that F � N , f =PF2F

�F � �F � f .

Note that the commuting properties seems to play important role not only for

Hom-functors. For example coherent functors introduced in [13] are characterized

in the module categories in [36, Lemma 1] as exactly those covariant functors which

commute with direct limits and direct products. The result was extended to locally

�nitely presented categories in [66, Chapter 9]. Commuting properties of covariant

Ext1-functors are studied in [24, 91, 50, 8, 86].


The defect functor Dev� = CokerHom(�;�) of a morphism � is a natural gen-

eralization of both covariant Hom and Ext1 functors in an arbitrary locally �nitely

presented abelian category.

If � : L! P is a homomorphism in C, then we have the following examples [23,

Example 2]:

(1) If C is abelian, P is projective and � a monomorphism, then Def�(�) is

canonically equivalent to Ext1(P=�(L);�).

(2) If P = 0, then Def�(�) is canonically equivalent to Hom(L;�).

(3) If � is an epimorphism and � : K ! L is the kernel of � then Def�(�)

represents the covariant defect functor associated to the exact sequence

0! K�! L

�! P ! 0.

(4) If R is a unital ring, C = Mod-R, and L and P are �nitely generated and

projective then Def�(R) represents the transpose of P=�(L).

Furthermore, the following criteria are known for an arbitrary homomorphism

� : L! P [23, Proposition 9]:

(1) Suppose that P is a compact object. The functor Def� commutes with

direct sums if and only if L is a compact object.

(2) Suppose that P is a �nitely generated object. The functor Def� commutes

with direct unions if and only if L is �nitely generated.

(3) Suppose that P is a �nitely presented object. Then Def� commutes with

direct limits if and only if L is �nitely presented.

As an analogue of [36, Lemma 1] in the case of direct unions the following result

can be proven:

Theorem 1.22. [23, Theorem 10] A functor F : C ! Ab commutes with respect

direct products and direct unions if and only if it is naturally isomorphic to a defect

functor Def� associated to a homomorphism � : L ! P with L and P �nitely

generated.

As an consequence we obtain for any homomorphism � : L ! P between pro-

jective modules equivalence of the three following properties:

(1) Def� commutes with direct sums,

(2) Def� commutes with direct unions

(3) Def� commutes with direct limits

Moreover, if these conditions are valid, then Def�(R) is a �nitely presented left

R-module [23, Proposition 11]. Let �J denote the canonical projection. Then the

commuting of Def� with a direct sum of objects can be characterize in the following

way:

Theorem 1.23. [23, Theorem 24] If � : L! P is a homomorphism and (Mi; i 2 I)

a family of objects, the following conditions are equivalent:

(1) Def� commutes with the direct sum of (Mi; i 2 I),


(2) for every f 2 Hom(L;L

i2I Mi) there exist �nite subset F � I, and g 2

Hom(P;L

i2InF Mi) such that �InF f = g�.

If � is a cardinal less than the �rst !-measurable cardinal and Def� commutes

with countable direct sums then Def� commutes with direct sums of � objects

[23, Proposition 26]. Thus in the constructible universe Def�(�) commutes with

countable direct sums if and only if Def�(�) commutes with all direct sums and,

in particular, for each M 2 C Ext1C(M;�) commutes with countable direct sums if

and only if Ext1C(M;�) commutes with all direct sums.

2. Semiartinian rings

Recall that a moduleM is semiartinian provided each non-zero factor ofM con-

tains a simple submodule and a ring R is right semiartinian if RR is a semiartinian

module. Of course, a right semiartinian ring can be characterized by the module

class conditions such that

(1) every module is semiartinian, or

(2) every non-zero module contains a simple submodule.

However the class of all semiartinian ring can be easily described by both ring-

theoretical and categorical conditions, it seems to interesting the question how the

structure of a semiartinian ring re ects some additional condition such as steadiness

or strongly steadiness. This way of research motivates the de�nition the right socle

chain, which is the uniquely de�ned strictly increasing chain of ideals (S� j � �

�+1) in a right semiartinian ring R satisfying S�+1=S� = Soc(R=S�), S0 = 0 and

S�+1 = R.

2.1. History. The notion generalizes the notion of a right artinian ring, which can

be described precisely as a semiartinian ring with the socle chain of a �nite socle

length and �nitely generated slices Soc(R=S�). Moreover, by [17, Theorem P] every

non-zero module over a left perfect ring has a non-zero socle, hence every left perfect

ring is right semiartinian. Basic structural results about general semiartinian rings

are published in papers [26, 39, 48, 72, 85]. Furthermore, let us recall the important

construction presented in the paper [16]:

Proposition 2.1. [16, Proposition 4.7] Let � be an in�nite cardinal, K a �eld

and R a semiartinian K-algebra with primitive factors artinian and socle chain

(S� j � � � ) for each < �. Let R =L

<�R + K �Q <�R and put

� = sup <� � . If either � is limit or f j � = �g is in�nite, then:

(1)L

<� S� is the �-th member of the socle chain 8� � �.

(2) If each R is right semiartinian, then R is right semiartinian with socle

length � + 1.

(3) R has primitive factors artinian.


Note that if the algebras in the construction are supposed to be (abelian) reg-

ular, then the constructed ring R is so. Actually, the classical result claims that

commutative semiartinian rings are close to abelian regular ones:

Theorem 2.2. [72, Th�eor�eme 3.1, Proposition 3.2] Let R be a ring.

(1) R is left semiartinian if and only if J(R) is right T-nilpotent and R= J(R)

is left semiartinian.

(2) Let R be a commutative semiartinian ring. Then R= J(R) is abelian regular

and semiartinian.

Properties and constructions of semiartinan rings close to von Neumann regular

ones are studied in papers [18, 15, 39, 83, 43] while papers [2, 1] are focused to

correspondence between the class of semiartinian rings and other interesting classes

of rings de�ned by some property of module categories.

2.2. Results. The notion of a dimension sequence plays an important role in re-

search of regular semiartinian ring with primitive factors artinian. Nevertheless,

before the de�nition we need to formulate the following result:

Theorem 2.3. [83, Theorem 2.1] Let R be a right semiartinian ring and L = (S� j

� � �+1) the right socle chain of R. Then the following conditions are equivalent:

(1) R is regular and all right primitive factor rings of R are right artinian,

(2) for each � � � there are a cardinal ��, positive integers n��, � < ��,

and skew-�elds K��, � < ��, such that S�+1=S� �=L

�<��Mn�� (K��),

as rings without unit. The pre-image of Mn�� (K��) coincides with the �-

th homogeneous component of R=S� and it is �nitely generated as right

R=S�-module for all � < ��. Moreover, �� is in�nite if and only if � < �.

If (1) holds true, then R is also left semiartinian, and L is the left socle chain of

R.

Denote by R the class of all regular right semiartinian rings R such that all

(right) primitive factor-rings of R are (right) artinian. If R 2 R, then the family

D(R) = f(��; f(n�� ;K��) j � < ��g) j � � �g

collecting data from the previous theorem is said to be the dimension sequence of

R.

The dimension sequences of a regular semiartinian ring naturally re ects the

structure of single semisimple slices. Note that structural theory of the notion is

developed in [110, 112]. An application of combinatorial set theory [46, 92] allows

to prove necessary conditions satis�ed by this invariant:

Theorem 2.4. [110, Proposition 3.1 and Theorem 3.5] Let R 2 R be abelian

regular, Generalized Continuum Hypothesis holds and �; � be ordinals satisfying

� + � � �. Then jh�; �ij � 2�� . If cf(��) > max(j�j; !), then ��+� � ��.

Otherwise ��+� � �+� .


On the other hand, commutative regular semiartinian rings with a particular

given rank of slices of the socle chain can be constructed:

Theorem 2.5. [110, Theorem 5.1] Let � be an ordinal, K a �eld, and (��j � � �)

a family of cardinals satisfying for every � � � � � the conditions:

(a) �� +� if cf(��) = !, and �� otherwise,

(b) �� < ! i� � = �,

(c) jh�; �ij � ��.

Then there exists a commutative regular semiartinian K-algebra with dimension

sequence f(��; f(1;K��) j � < ��g) j � � �g where K�� = K for all � � � and

� < ��.

Furthermore, it is possible to generalize results on dimension sequences for a suit-

able subclass of regular right semiartinian rings R with primitive factors artinian,

namely, for those R satisfying the condition that every ideal which is �nitely gener-

ated as two-sided ideal is �nitely generated as right ideal. It is proved in the paper

[112, Theorem 3.4] that over these rings and under Generalized Continuum Hypoth-

esis it holds that ��+�(n) � ��(m), wheneverm � n and �; � are ordinals such that

�+� � � and cf(��(n)) > max(j�j; !), where f(��; f(n�� ;K��) j � < ��g) j � � �g

is dimension sequence and ��(n) = cardf� < ��j n�� ng).

3. Tall rings

A moduleM which contains a non-noetherian submodule N such that the factor

M=N is non-noetherian as well is studied �rst in the paper [84] under the term tall.

The notion of a right tall ring is de�ned in the same paper as a ring over which

every non-noetherian right module is tall. Note that this notion presents a "typical"

example of a ring described by a module-class property.

3.1. History. It is not hard to see that the class of all right tall rings is closed under

factors, �nite products, and Morita equivalence. Although in [84] is presented a

criterion of right tall rings using the notion of Krull dimension of all modules, an

existence a general ring-theoretic necessary and su�cient condition remains to be

an open problem.

Theorem 3.1. [84, Theorem 2.7] The following statements are equivalent for a

ring R:

(1) R is right tall,

(2) every non-noetherian module has a proper non-noetherian submodule,

(3) every module with Krull dimension is noetherian.

Since every maximal submodule of a non-noetherian module is non-noetherian,

the condition (2) implies that every right max ring, over which every nonzero right

module contains a maximal submodule, is necessarily right tall [31, p. 31]. Never-

theless, the following example shows that the revers implication does not hold.


Example 3.2. [76, Example 3.2] Put I =P

i x2iF [X] and R = F [X]=I for a �eld

F and an in�nite countable set of variables X = fx1; x2; : : : g. Let Xi = xi + I

and de�ne an ideal J =P

iXiR. Then J is a nil ideal, since X2i = 0 and R is

commutative. As R=J �= F , R is tall ring by [76, Lemma 3.1]. Moreover, J is a nil

maximal ideal of R, thus it is the Jacobson radical of R. Since X1 � � � � �Xn 6= 0 for

every n, J is not T-nilpotent, hence R is not a max ring.

No general ring-theoretical criterion characterizing max rings neither correspon-

dence between the classes of all tall and all max rings is known. Nevertheless, max

rings are studied by many authors from various points of view and with di�erent

motivations [25, 27, 31, 47, 53, 63, 97]. Among another results let us recall several

classical module-theoretical necessary and su�cient conditions:

Theorem 3.3. [47, 53, 63] The following conditions are equivalent for a ring R:

(1) R is a right max ring;

(2) R=J(R) is a right max ring and J(R) is right T-nilpotent,

(3) every non-zero submodule of injective envelops E(S) contains a maximal

submodule for every simple module S,

(4) there is a cogenerator for the category of right modules whose every non-zero

submodule contains a maximal submodule.

Much more is known about both commutative max rings and commutative tall

rings. The most important fact from our point of view is ring-theoretical criteria

of commutative max ring:

Theorem 3.4. [47, 53, 63] The following conditions are equivalent for a commu-

tative ring R:

(1) R is a max ring;

(2) R=J(R) is a regular ring and J(R) is left T-nilpotent,

(3) the localization at any maximal ideal of R is a max ring,

(4) the localization at any maximal ideal of R is a perfect ring.

3.2. Results. For description of commutative tall rings are very useful to formulate

necessary structural condition of non-tall rings:

Theorem 3.5. [76, Theorem 2.6] Let R be a commutative non-tall ring. Then

there exists a maximal ideal I and a sequence of ideals I = J1 � J2 � : : : such that

(1) IJi � Ji+1 for each i,

(2) R=Ji is artinian for each i,

(3)Ti Ji is a prime ideal,

(4) R=Tn I

n is not a tall ring.

Note that ideals Jn from the previous theorem cannot be replaced by the powers

Jn1 in general [76, Example 3.7]. On the other hand, if R is tall, then for every


non-idempotent maximal ideal I such that R=Ii is artinian for each i, the inter-

sectionTj I

j is not a prime ideal [76, Proposition 2.9]. As the consequence can be

formulated the following criterion:


commutative ring R:

(1) R is not tall,

(2) there exists a non-noetherian artinian module,

(3) there exists an artinian module M , elements x 2 R and mj 2M such that

mj+1x = mj and mj+1 =2 mjR for each j and M =SjmjR.

(4) there exists a sequence of ideals Jj of R and elements xj 2 R such that R=Jj

is artinian, Jj+1 ( Jj, xjr 2 Jj+1 i� r 2 Jj and the length of Sj(R=Jj) is

equal to the length of Sj(R=Jk) for each j � k < !.

Finally note that the previous criterion can be expressed in a very simple form

in the case of a commutative noetherian ring R, namely, R is tall if and only if R

is artinian [76, Theorem 2.10].

4. Retractability and coretractability

Both the central notions of this section, i.e. completely coretractable and mod-

retractable rings present examples of rings naturally determined by a categorical

property. A module M such that its every nonzero submodule contains a nonzero

endomorphic image of M is called retractable and, dually, M is called coretractable

if there exists a nonzero homomorphism of M=K to M for every proper submodule

K � M . For example each �nitely generated module over commutative ring is

retractable [41].

4.1. History. The importance of the notions has emerged in research of Baer mod-

ules [81, 82], endomorphism rings of nonsingular modules [61, 62], compressible

modules [87, 89] and module lattices [51, 101]. The works [10, 41, 52] are devoted

to rings over which every module is retractable or coretractable.

Main results of [10] describe rings over which every right module is coretractable,

such rings are called right completely coretractable. Dually, a ring R is said to be

right mod-retractable provided every right R-module is retractable.

It is proved in [111, Theorem 2.4] that a ring is right (left) completely core-

tractable if and only if it is isomorphic to a �nite product of matrix rings over

right and left perfect rings. Furthermore, every cyclic right and left R-module is

coretractable [111, Proposition 3.2].

The papers [41, 52] started to study mod-retractable rings. Note that the class of

mod-retractable rings is closed under Morita equivalence, factorization, and �nite

products [41]. Moreover it is known that any right mod-retractable ring is an

example of a right max ring [64, Theorem 3.3].


4.2. Results. However mod-retractable rings are precisely rings such that all their

torsion theories are hereditary, the general ring-theoretical criterion of mod-retract-

ability is not know. The characterization is available only for several particular

classes of rings and for all commutative rings:

Theorem 4.1. [64, Theorem 3.3] Let R be a left perfect ring. Then R is right

mod-retractable if and only if R �=Qi�kMni(Ri) for a system of a local rings Ri,

i � k, which are both left and right perfect.

As an easy consequence can be shown that every commutative perfect ring is

mod-retractable.

A similar criterion is proved for the class of right noetherian rings:

Theorem 4.2. [64, Theorem 3.3] Let R be a right noetherian ring. Then R is right

mod-retractable if and only if R �=Qi�kMni(Ri) for a local right artinian rings Ri.

As it was mentioned a criterion of mod-retractability is known for the class of

commutative rings:

Theorem 4.3. [64, Theorem 3.10] Let R be a commutative ring. Then R is mod-

retractable if and only if R is semiartinian.

Finally note that from the previous result immediately follows that every com-

mutative semiartinian ring is necessarily mod-retractable.

5. RM rings

First recall that a module M satis�es the restricted minimum condition if for

every essential submodule N of M , the factor M=N is artinian. The class of all

modules satisfying the restricted minimum condition is well-known to be closed

under submodules, factors as well as �nite direct sums. Note that a semiartinian

module M satis�es the restricted minimum condition if and only if M= Soc(M) is

artinian.

5.1. History. A ring R is called a right RM-ring if RR satis�es restricted minimum

condition as a right module. Obviously, the class of all right RM-rings contains all

right artinian rings and principal ideal domains. This observation partially explains

the historical motivation of research of these rings. Structure theory of RM-rings

and domains was studied in the papers [28, 29, 34, 37, 73]. Among others let us

recall the following result:

Theorem 5.1. [34, Theorem 1] Let R be a noetherian domain. Then R has Krull

dimension 1 if and only if it is an RM-domain

However the de�nition of RM-rings has a ring-theoretical nature (it actually deals

with cyclic modules), the correspondence between RM-rings and the classes of rings

studied above is clari�ed in the context of results of the paper [6], which is devoted


to structure research of classes of torsion modules over RM-domains. Namely, it

seems to be interesting question here whether there exists nice categorical property

which is equivalent to the ring-theoretical de�nition.

5.2. Results. Ring-theoretical results for non-commutative as well as commutative

rings are proved in [113]. Recall a useful technical result which consists of several

necessary conditions of modules over general RM-ring where E(M) denotes the

injective envelope of a module M :

Theorem 5.2. [113, Theorem 2.11] Let R be a right RM-ring and M a right R-

module.

(1) If M is singular, then M is semiartinian.

(2) E(M)=M is semiartinian.

(3) If M is semiartinian, then E(M) is semiartinian. In particular, E(S) is

semiartinian for every simple module S.

As an consequence it can be obtained the observation for a right RM-ring R

that R is a nonsingular ring of �nite Goldie dimension whenever Soc(R) = 0 [113,

Theorem 2.12].

For a semilocal RM-rings can be proved the following criterion:

Theorem 5.3. [113, Theorem 2.17] Let R be a semilocal RM-ring and Soc(R) = 0.

Then the following conditions are equivalent:

(1) R is noetherian,

(2) J(R) is �nitely generated,

(3) the socle length of E(R=J(R)) is at most !.

Recall characterization of commutative RM-domains from the paper [6] which

motivates are research:

Theorem 5.4. [6, Theorem 6 and 9] The following conditions are equivalent for a

commutative domain R:

(1) R is an RM-ring,

(2) M = �P2Max(R)M[P ] for all torsion modules M ,

(3) R is noetherian and every non-zero (cyclic) torsion R-module has an es-

sential socle,

(4) R is noetherian and every self-small torsion module is �nitely generated.

The most important result of the article [113] describes commutative RM-ring

in the language of module categories which generalizes the previous result:


commutative ring R:

(1) R is an RM-ring,

(2) M = �P2Max(R)M[P ] for all singular modules M ,


(3) R=Soc(R) is Noetherian and every self-small singular module is �nitely

generated.

The question whether a similar result is valid in the non-commutative case re-

mains open.

6. Conclusion

Let us summarize the contribution of the present thesis:

(1) We describe the structure of rings belonging to classes determined by a

particular categorical property. Namely, in [22], [105], [76], [64], and [65]

respectively are characterized several subclasses of right strongly steady,

steady, tall, mod-retractable, and RM-rings. The structural theory of

abelian regular semiartinian rings is developed in [110].

(2) We answer several structural question on classes of compact objects, in

particular, [106] is devoted to closure properties of the class of all self-

small modules and [102] determines test modules for the class of all small

modules.

(3) We contribute to the study of commuting properties of functors in [23].

References

[1] A.N. Abyzov: Generalized SV-rings of bounded index of nilpotency. Russ. Math. 55, No. 12,1-10 (2011).

[2] A.N. Abyzov: Regular semi-Artinian rings. Russ. Math. 56, No. 1, (2012), 1-8.[3] J. Ad�amek, J. Rosick�y: Locally presentable categories and accessible categories, London

Math. Soc. Lec. Note Series 189 (1994).[4] J. Ad�amek, J. Rosick�y, E. Vitale, Algebraic Theories: A Categorical Introduction to General

Algebra, Cambridge Tracts in Mathematics 184, Cambridge University Press, (2010).[5] U. Albrecht Quasi-decompositions of abelian groups and Baer's Lemma, Rocky Mount. J.

Math., 22 (1992), 1227{1241.[6] Albrecht U., Breaz, S.: A note on self-small modules over RM-domains, J. Algebra Appl.

13(1) (2014), 8 pages.[7] Albrecht, S. Breaz, P. Schultz: The Ext functor and self-sums, Forum Math. 26 (2014),

851{862.[8] U. Albrecht, S. Breaz, P. Schultz: Functorial properties of Hom and Ext, Contemporary

Mathematics 576 (2012), 1{15.[9] U. Albrecht, S. Breaz, W. Wickless: The �nite quasi-Baer property. J. Algebra 293 (2005),

no. 1, 116.[10] B.Amini, M.Ershad and H. Sharif, Coretractable modules, J. Aust. Math. Soc. 86 (2009),

No. 3, 289{304.[11] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules. 2nd edition, Springer,

New York, 1992.[12] D. M. Arnold and C. E. Murley: Abelian groups, A, such that Hom(A;�) preserves direct

sums of copies of A, Paci�c J. Math., 56, (1975), 7{21.[13] M. Auslander: Coherent functors, Proc. Conf. Categor. Algebra, La Jolla 1965, (1966), 189{

231.[14] M. Auslander, I. Reiten, S.O. Smal�o: Representation theory of Artin algebras, Cambridge

Studies in Advanced Mathematics 36. Cambridge: Cambridge University Press, 1995.[15] Baccella, G.: On C-semisimple rings. A study of the socle of a ring, Comm. Algebra, 8(10)

(1980), 889-909.[16] Baccella, G.: Semiartinian V-rings and semiartinian von Neumann regular rings. J. Algebra

173 (1995), 587{612.


[17] Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans.Am. Math. Soc. 95 (1960), 466{488.

[18] H. Bass: Algebraic K-theory, Mathematics Lecture Note Series, New York-Amsterdam: W.A.Benjamin, 1968.

[19] R. Bieri, B. Eckmann: Finiteness properties of duality groups, Commentarii Math. Helvet.49 (1974), 74{83.

[20] S.Breaz: Self-small abelian groups as modules over their endomorphism rings, Comm. Algebra31 (2003), no. 10, 4911{4924.

[21] S. Breaz The quasi-Baer-splitting property for mixed abelian groups, J. Pure Appl. Algebra,191 (2004), 75{87.

[22] S. Breaz, J. �Zemli�cka: When every self-small module is �nitely generated. J. Algebra 315/2(2007), 885{893.

[23] S. Breaz, J. �Zemli�cka: The defect functor of homomorphisms and direct unions, Algebr.Represent. Theor. 19/1 (2016), 181{208.

[24] K. S. Brown: Homological criteria for �niteness, Commentarii Math. Helvet. 50 (1975), 129{135.

[25] V. P. Camillo, On some rings whose modules have maximal submodules, Proc. Amer. Math.Soc. 50 (1975), 97{100.

[26] Camillo V.P., Fuller, K.R.: On Loewy length of rings. Pac. J. Math. 53 (1974), 347{354 .[27] S. Charalambides, Stelios; J. Clark, Max modules relative to a torsion theory, J. Algebra

Appl. 7 (2008), No. 1, 21-45.[28] A.W.Chatters: The restricted minimum condition in Noetherian hereditary rings, J. Lond.

Math. Soc., II. Ser. 4 (1971), 83-87 .[29] A.W.Chatters, C.R. Hajarnavis: Rings with Chain Conditions; Pitman Advanced Publishing

44; Boston, London, Melbourne (1980).[30] S. U. Chase: Direct products of objects, Trans. Amer.Math. Soc. 97 (1960), 457-473.[31] J. Clark, On max modules, Proceedings of the 32nd Symposium on Ring Theory and Repre-

sentation Theory, Tokyo 2000, 23{32.[32] R. Colpi and C. Menini, On the structure of �-modules, J. Algebra 158, 1993, 400{419.[33] R. Colpi and J. Trlifaj, Classes of generalized �-modules, Comm. Algebra 22, 1994, 3985{

3995.[34] Cohen, I.S.: Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950),

27-42.[35] J. Cornick, I. Emmanouil, P. Kropholler, O. Talelli: Finiteness conditions in the stable module

category, Advances in Mathematics, 260 (2014), 375{400.[36] W. Crawley-Boevey: In�nite-dimensional modules in the representation theory of �nite-

dimensional algebras, Canadian Math. Soc. Conf. Proc., 23 (1998), 29{54.[37] P. Dan, D. V. Huynh: On rings with restricted minimum condition, Arch. Math. 51(4)(1988),

313-326 .[38] R. Dimitric: Slender modules over domains, Comm. Algebra 11 (1983), no. 15, 1685{1700.[39] Dung N.V., Smith, P.F.: On semi-artinian V -modules. J. Pure Appl. Algebra 82 (1992),

27{37.[40] J. Dvo�r�akk: On products of self-small abelian groups, Stud. Univ. Babe-Bolyai Math. 60

(2015), no. 1, 1317.[41] S�. Ecevit and M. T. Ko�san, On rings all of whose modules are retractable, Arch. Math. (Brno)

45 (2009), 71{74.[42] K. Eda: Slender modules, endo-slender abelian groups and large cardinals, Fund. Math. 135

(1990), no. 1, 5{24.[43] P.C. Eklof, K.R. Goodearl and J. Trlifaj, Dually slender modules and steady rings, Forum

Math., 1997, 9, 61{74.[44] P. Eklof, A. Mekler: Almost Free objects: Set-theoretic methods, Revised edition , North-

Holland Mathematical Library vol. 65 (2002).[45] R. El Bashir,T. Kepka, P. N�emec, Modules commuting (via Hom) with some colimits:

Czechoslovak Math. J. 53 (2003), 891{905.[46] Erd}os P., Rado, R.: A partition calculus in set theory. Bull. Amer. Math. Soc., 62 (1956),

427{489.[47] C. Faith, Rings whose modules have maximal submodules, Publ. Mat. 39 (1995), 201{214.[48] L. Fuchs: In�nite Abelian Groups, Vol I, Academic Press (1970).


[49] K. R. Goodearl, Von Neumann Regular Rings, London, 1979, Pitman, Second Ed., Mel-bourne, FL, 1991, Krieger.

[50] R. G�obel, J. Trlifaj: Endomorphism Algebras and Approximations of objects, Expositions inMathematics 41, Walter de Gruyter Verlag, Berlin (2006).

[51] A. Haghany, M. R. Vedadi, Study of semi-projective retractable modules, Algebra Colloq. 14(2007), no. 3, 489{496.

[52] A. Haghany, O. A. S. Karamzadeh, M. R. Vedadi, Rings with all �nitely generated modulesretractable, Bull. Iranian Math. Soc. 35 (2009), no. 2, 37{45.

[53] Y. Hirano, On rings over which each module has a maximal submodule, Comm.Algebra 26

(1998), 3435{3445.[54] T. Head, Preservation of coproducts by HomR(M,-), Rocky Mt. J. Math. 2, (1972), 235{237.[55] R. Hartshorne: Coherent functors, Adv. Math. 140 (1998), 44{94.[56] M. H�ebert: What is a �nitely related object, categorically?, Applied Categorical Structures,

21 (2013), 1{14.[57] Jech, T.: Set theory. The third millennium edition, revised and expanded.. Springer Mono-

graphs in Mathematics, Springer, Berlin - Heidlberg 2003.[58] C. U. Jensen, H. Lenzing: Model theoretic algebra: with particular emphasis on �elds, rings,

modules, Algebra, Logic and Applications 2, Gordon and Breach Science Publishers, NewYork, (1989).

[59] S. M. Khuri: Endomorphism rings and lattice isomorphisms, J. Algebra, 56(1979), 401-408.[60] S. M. Khuri: Endomorphism rings of nonsingular modules, Ann. Sci. Math. Qu., 4(1980),

145-152.[61] S. M. Khuri, Nonsingular retractable modules and their endomorphism rings, Bull. Aust.

Math. Soc. 43 (1991), No.1, 63{71[62] S. M. Khuri, The endomorphism ring of a nonsingular retractable module, East-West J.

Math. 2 (2000), 161{170.[63] L. A. Koifman, Rings over which every module has a maximal submodule, Mat. Zametki 7

(1970), 359{367; (transl.) Math. Notes 7 (1970), 215{219.[64] M. T. Koan, J. �Zemli�cka: Mod-retractable rings, Commun. Algebra 42/3 (2014) 9981010.[65] M. T. Koan, J. �Zemli�cka: On modules and rings with restricted minimum condition, Colloq.

Math., 140,1 (2015), 75-86.[66] H. Krause: The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), no. 707.[67] T.Y. Lam Lectures on Modules and Rings, Springer-Verlag, New York, 1999.[68] H. Lenzing: Endlich pr�asentierbare Moduln, Arch. Math. (Basel) 20 (1969), 262{266.[69] B. Goldsmith and O. Kolman: On cosmall Abelian groups, J. Algebra, 317, (2007), 510{518.[70] S. MacLane: Categories for the working mathematician, Graduate texts in mathematics 5,

Springer-Verlag, 1998.[71] Jun-Ichi Miyachi, Compact objects in stable module categories Arch. Math. 89 (2007), 47-51.[72] C. N�ast�asescu and N. Popescu, Anneaux semi-artiniens, Bull. Soc. Math. France 96 (1968),

357{368.[73] A.J. Ornstein: Rings with restricted minimum condition, Proc. Am. Math. Soc. 19 (1968),

1145{1150.[74] J.D.O'Neill: Slender modules over various rings, Indian J. Pure Appl. Math. 22 (1991), no.

4, 287{293.[75] J.L. G�omez Pardo, G. Militaru and C. N�ast�asescu, When is HOMR(M;�) equal to

HomR(M;�) in the category R-gr?, Comm. Algebra, 22, (1994), 3171-3181.[76] T. Penk, Jan �Zemli�cka, Commutative tall rings, J. Algebra Appl., 13/4 (2014)[77] G. Puninski: Pure projective modules over an exceptional uniserial ring, St. Petersburg Math.

J. 13(6) (2002), 175{192.[78] R. Rentschler: Die Vertauschbarkeit des Hom-Funktors mit direkten Summen, Dissertation,

Lud-wig-Maximilians-Univ., Munich, 1967.[79] R. Rentschler: Sur les objects M tels que Hom(M;�) commute avec les sommes directes, C.

R. Acad. Sci. Paris Sr. A-B 268 (1969), 930{933.[80] S. T. Rizvi and C. S. Roman: Baer and quasi-Baer modules, Comm. Algebra, 32(1)(2004),

103-123.[81] S.T. Rizvi, C. S. Roman, On K-nonsingular modules and applications, Commun. Algebra 35

(2007), 2960{2982.[82] S.T. Rizvi, C. S. Roman, On direct sums of Baer modules, J. Algebra 321 (2009), 682{696.


[83] P.R�u�zi�cka , J. Trlifaj and J. �Zemli�cka: Criteria of steadiness, Abelian Groups, Module Theory,and Topology, New York 1998, Marcel Dekker, 359{372.

[84] B. Sarath, Krull dimension and noetherianness, Illinois J. Math. 20 (1976), 329{335.[85] Salce, L., Zanardo, P.: Loewy length of modules over almost perfect domains. J. Algebra 280

(2004), 207{218.[86] P. Schultz: Commuting Properties of Ext, J. Aust. Math. Soc., 94 (2013), no. 2, 276{288.[87] P. F. Smith, Modules with many homomorphisms, J. Pure Appl. Algebra 197 (2005), 305{

321.[88] P. F. Smith, Compressible and related modules, Abelian groups, rings, modules, and homo-

logical algebra, 295313, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, BocaRaton, FL, 2006.

[89] P. F. Smith, M. R. Vedadi, Submodules of direct sums of compressible modules, Comm.Algebra 36 (2008), no. 8, 3042{3049.

[90] Bo Stenstr�om, Rings of Quotients, Berlin, 1975, Springer-Verlag.[91] R. Strebel: A homological �niteness criterion, Math. Z. 151 (1976), 263{275.[92] Tarski, A.: Sur la d�ecomposition des ensembles en sous-ensembles presque disjoints. Fundam.

Math. 12 (1928), 188{205.[93] J. Trlifaj Almost �-modules need not be �nitely generated, Comm. Algebra, 21 (1993), 2453{

2462.[94] J. Trlifaj: Strong incompactness for some nonperfect rings, Proc. Amer. Math. Soc. 123

(1995), 21{25.[95] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents, in Abelian groups

and objects (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995, 467{473.[96] J. Trlifaj: Whitehead test modules, Trans. Amer. Math. Soc. 348 (1996), no. 4, 15211554.[97] A. A. Tuganbaev, Rings whose nonzero modules have maximal submodules, J. Math. Sci.

109, 1589-1640.[98] Williams, N.H.: Combinatorial Set Theory. North-Holland, Amsterdam 1977.[99] R. Wisbauer: Foundations of object and Ring Theory, Gordon and Breach, Reading, (1991).[100] J. M. Zelmanowitz, Correspondences of closed submodules, Proc. Am. Math. Soc. 124

(1996), No.10, 2955-2960.[101] Zhengping Zhou, A lattice isomorphism theorem for nonsingular retractable modules, Can.

Math. Bull. 37 (1994), No.1, 140{144.[102] J. �Zemli�cka: Steadiness is tested by a single module, Contemporary Mathematics, 273

(2001), 301{308.[103] J. �Zemli�cka !1-generated uniserial modules over chain rings, Comment. Math. Univ. Car-

olinae, 45 (2004), 403{415.[104] J. �Zemli�cka: Classes of dually slender modules, Proceedings of the Algebra Symposium,

Cluj, 2005, Editura Efes, Cluj-Napoca, 2006, 129{137.[105] J. �Zemli�cka: Steadiness of regular semiartinian rings with primitive factors artinian,

J.Algebra, 304 (2006), 500{509.[106] J. �Zemli�cka: When products of self-small modules are self-small, Commun. Algebra 36/7

(2008), 2570-2576.[107] J. �Zemli�cka: Large small modules over von Neumann regular rings, In: Proceedings of

the International Conference on Modules and Representation Theory, Presa Univ. Clujean�a,Cluj-Napoca, 2009, 211{220.

[108] J. �Zemli�cka: Steadiness of polynomial rings, Algebra Discrete Math. 10/2 (2010), 107-117.[109] J. �Zemli�cka: Small modules over abelian regular rings, Commun. Algebra 40/7 (2012),

2485{2493.[110] J. �Zemli�cka: Socle chains of abelian regular semiartinian rings. J. Pure Appl. Algebra 217/6

(2013), 1018{1025.[111] J. �Zemli�cka: Completely coretractable rings, Bull. Iran. Math. Soc. 39/3 (2013), 523-528.[112] J. �Zemli�cka: On socle chains of semiartinian rings with primitive factors artinian,

Lobachevskii J. Math., 37/3 (2016) 316{322,.[113] J. �Zemli�cka and J. Trlifaj: Steady ideals and rings, Rend. Sem. Mat. Univ. Padova, 98

(1997), 161{172.

Chapter 2

Self-small modules and strongly

steady rings

The main goal of this chapter is to describe structural properties of classes of selfsmall modules. The text consists of two papers:

A. Simion Breaz, Jan �Zemli�cka, When every self-small module is �nitely gen-

erated, J. Algebra 315/2 (2007), 885{893.

B. Jan �Zemli�cka, When products of self-small modules are self-small, Commun.Algebra 36/7 (2008), 2570{2576.

26

A. WHEN EVERY SELF-SMALL MODULE IS FINITELY

GENERATED

SIMION BREAZ AND JAN �ZEMLI�CKA

Abstract. The aim of this paper is to give necessary and su�cient condi-tions for rings for which every right self-small module is �nitely generated. Itis proved that: semi-simple rings, commutative perfect rings and right non-singular �-extending rings have this property; a right nonsingular semi-primering has the property if and only if it is semi-simple; a commutative noetherianring has the property if and only if it is artinian.

1. Introduction

Let R be a unital ring. A right R-module M is called self-small if the covariant

functor Hom(M;�) commutes with direct sums of copies of M . This notion was

introduced by Arnold and Murley in [2] in order to study some generalization of

a well known Baer's Lemma, [10, Proposition 86.5]. Meanwhile, these modules

were useful in the study of a large variety of properties as splitting properties, [1]

and [5], properties for homomorphisms of graded modules, [11]. Maybe the most

important utility of self-small modules is in the study of representable equivalences

between subcategories of module categories in connection with tilting theory. In

this context they are viewed as generalizations for �nitely generated modules and

they are used in the de�nition of ?-modules, [6]. In [18] it is proved that every

?-module is �nitely generated. However, the property \�nitely generated" is not

valid for some generalizations of ?-modules, [19]. These two results had lead to the

following problem: \Characterize the rings for which every small right module is

�nitely generated". These rings are called right steady and they are studied in [7],

[9], [20], [21], [22]. We recall that a right R-module M is small if the covariant

functor Hom(M;�) commutes with arbitrary direct sums.

In the present paper we consider the problem: \Characterize the rings for which

every self-small right module is �nitely generated". We call these rings right strongly

steady. The class of right steady rings is larger than the class of right strongly steady

rings. To see this it is enough to consider the ring Z of all integers. This ring is

right steady but it is not a right strongly steady ring. For example Q is a self-small

Z-module which is not �nitely generated.

2000 Mathematics Subject Classi�cation. 16D10(16P20;16P70).Key words and phrases. Self-small module, strongly steady ring, semi-simple ring, right non-

singular ring, artinian ring, noetherian ring.The �rst author was supported by the grant CEEX ET47/2006. The second author was

supported by grant GAUK 448/2004/B-MAT and by the research project MSM 0021620839.

27

28 SIMION BREAZ AND JAN �ZEMLI�CKA

In the second section we present some basic facts. It is proved that a ring is right

strongly steady if it is semi-simple (Proposition 2.5) or perfect and commutative

(Theorem 2.9). We mention here that in [7, Corollary 1.6] it is proved that every

right perfect ring is right steady. This is not the case for the strongly steady

property. We show in Example 3.11 that a right artinian ring which is not right

strongly steady does exist.

In Section 3 we consider right non-singular rings which are right strongly steady.

The study of this case is in fact a part of a more general problem in module theory

which asks to �nd correspondences between properties of a ring S and a unital

subring R � S such that S is �nitely generated as a right R-module, [4], [13,

Section 3]. If R is a right strongly steady right non-singular ring then it is right

artinian (Theorem 3.9). Moreover, a right nonsingular ring which is commutative

or semi-prime is right strongly steady if and only if it is semi-simple (Theorem 3.8

and Theorem 3.12). In Example 3.13 we proved that there exist right nonsingular

strongly steady rings which are not semi-simple. Finally it is shown in Proposition

3.16 that every right noetherian strongly steady ring is right artinian. Using these

we prove in Theorem 3.17 that a commutative noetherian ring is strongly steady if

and only if it is artinian.

In this paper every ring is associative and unital. If R is a ring, then J denotes

the Jacobson radical of R. A module (or R-module) means right module over a

ring R. All unexplained notions and notations can be founded in [13], [14] and [16].

2. Basic facts

We start with some closure properties for the class of right strongly steady rings.

Lemma 2.1. Strongly steadiness is preserved by factorization.

Proof. Let R be a right strongly steady ring and I an ideal of R. The ring

R=I is right strongly steady as a consequence of the equality HomR(M;N) =

HomR=I(M;N) for all right R=I-modules M and N . �

Lemma 2.2. Every in�nite product of rings contains a self-small non-�nitely gen-

erated ideal.

Proof. By [17, Lemma 2.4] any in�nite product of rings contains non-�nitely gen-

erated ideal which is small as a right and a left ideal. �

Lemma 2.3. Let Ri, i 2 I be rings.Qi2I Ri is right strongly steady if and only if

I is �nite and Ri is right strongly steady for every i 2 I.

Proof. ()) By Lemma 2.2, the set I is �nite. Any ring Ri is right strongly steady

since it is a factor ofQi2I Ri.

(() Let M be a self-small module. Then every right R-module M is isomorphic

to a �nite direct product M �=Qi2I Mi, where each Mi is a right Ri-module. It is

not hard to observe that M is a self-small right R-module if and only if every Mi

A. WHEN EVERY SELF-SMALL MODULE IS FINITELY GENERATED 29

is self-small as an Ri-module. Since Mi is �nitely generated for each i 2 I, M is

�nitely generated as well. �

Since Morita equivalences preserve (and re ect) �nitely generated modules as

well as self-small modules, we deduce immediately the following result.

Lemma 2.4. The property \to be right strongly steady" is preserved by Morita

equivalences.

Proposition 2.5. Every semi-simple ring is right strongly steady.

Proof. It follows immediately from Lemma 2.3 and Lemma 2.4. �

Lemma 2.6. Let R be a right perfect ring. If 0 6=M is a right R-module, then M

has a non-zero simple quotient.

Proof. The Jacobson radical of R is right T -nilpotent, hence MJ 6= M by [14,

Theorem 23.16]. Moreover, M=MJ can be viewed as a right R=J-module. Since

R=J is semi-simple, it has a non-zero simple summand (as an R=J-module). But

every simple R=J-module is simple as an R-module, and the proof is complete. �

For the convenience of the reader we recall a characterization of self-small mod-

ules which was proved in [2, Proposition 2.1].

Proposition 2.7. A right R-module M is not self-small if and only if there exists

a countable chain M0 � M1 � � � � � Mn � : : : ; n < ! of submodules of M such

that M =Sn<!Mn and for every n < ! there exists a non-zero endomorphism

fn :M !M such that fn(Mn) = 0.

Remark 2.8. We can suppose that the chain of submodules in the previous propo-

sition is strictly increasing. We recall that a right R module is not small if and only

if it is a union of a strictly increasing countable chain of submodules.

Proposition 2.9. Every commutative perfect ring is (right) strongly steady.

Proof. A commutative perfect ring is product of �nitely many local perfect rings,

hence it is enough to prove the property for the local case.

Let R be a local commutative perfect ring and S = R=J , the simple R-module.

Fix an R-module M which is not �nitely generated. Since R is steady by [7,

Corollary 1.6], M is not a small module. Therefore, we have a strictly increasing

chain of submodules Mn ( Mn+1 ( M such that M =Sn<!Mn. If n < !, since

M=Mn 6= 0, there exists an epimorphism M=Mn ! S by Lemma 2.6. But M has

a submodule isomorphic to S as a consequence of Bass Theorem. Therefore, for

every n < ! there is a non-zero homomorphism fn :M !M such that f(Mn) = 0,

which proves M is not self-small. �


3. Right nonsingular and noetherian rings

Suppose that G is a Gabriel topology on the ring R and M is an R-module. We

will denote, as in [16], by MG the module of quotients of M with respect G.

Lemma 3.1. Let R be a right non-singular ring, G a Gabriel topology on R and

N a G-torsion-free right R-module. If K is an R-submodule of N and M is the

RG-submodule of NG which is generated by K then M=K is a G-torsion R-module.

Proof. We consider KG as a submodule of NG and we construct M as a submodule

of KG. Then M=K is a submodule of the G-torsion R-module KG=K, hence M=K

is G-torsion since the torsion theory induced by G is hereditary. �

Recall a useful observation made in [8, Lemma 12.4] and [16, Exercise IX.15]):

Lemma 3.2. Let R be a ring and G a Gabriel topology on R. If M and N are

RG-modules such that N a G-torsion-free as an R-module then the canonical map

HomRG(M;N)! HomR(M;N) is an isomorphism.

Lemma 3.3. Let R be a right non-singular ring and G a Gabriel topology on R.

(1) If M is a G-torsion-free small R-module then MG is self-small as an R-

module.

(2) If M is a self-small RG-module which is G-torsion-free as an R-module

then M is self-small as an R-module.

Proof. (1) Let f :MG !M(I)G

be an R-homomorphism. SinceM is small, there ex-

ists a �nite subset F � I such that f(M) �M(F )G

. Therefore the R-homomorphism

f :MG=M !M(InF )G

, f(x+M) = �InF f(x) is well de�ned (�InF :M(I)G

!M(InF )G

denotes the canonical projection). But HomR(MG=M; (M(InF )G

) = 0 since MG=M

is G-torsion and MG is G-torsion free, hence f = 0. It follows that f(MG) �M(F )G

.

(2) If I is a set then M (I) is G-torsion-free as a right R-module, hence every

R-homomorphism f : M ! M (I) is an RG-homomorphism by Lemma 3.2, and it

follows that there exists a �nite subset F � I such that Im(f) �M (F ). �

As usual, we denote by Qmax the maximal right ring of quotients of a ring R.

Since Qmax is torsion-free over any nonsingular ring, we may conclude the following

consequence of Lemma 3.3 (1):

Corollary 3.4. If R is a right non-singular ring, then Qmax is a self-small R-

module.

For the proof of the next proposition we use an idea presented in [9, Theorem 2.8].

We also need some technical results.

For an arbitrary cardinal � we say that a moduleM is �-reducing if for every sub-

module N �M such that gen(N) � � there exists a �nitely generated submodule

F for which N � F �M .


Remark 3.5. Note that a �-reducing module is small by [9, Lemma 1.5], hence

self-small, for every in�nite cardinal �.

Lemma 3.6. Let R be a subring of a ring Q such that Q is �nitely generated as a

right R-module.

(1) Every non-�nitely generated !-reducing Q-module is non-�nitely generated

!-reducing as an R-module.

(2) If Q is simple regular which is not right artinian, every injective Q-module

is small as an R-module.

Proof. (1) An easy observation.

(2) Since Q is a non-artinian regular ring, it contains a set of non-zero orthogonal

idempotents fenjn < !g, i.e. we have a right idealL

n<! enQ � Q.

Now, we modify the proof of [7, Lemma 1.10]. Let P be an injective Q-module

and assume that P =Sn<! Pn for a strictly increasing chain of R-submodules

P0 ( P1 ( � � � ( Pn ( � � � . Note that PenQ = P becauseQ is a simple ring, so there

exists pn 2 P such that pnenQ * Pn. As P is Q-injective, the Q-homomorphism

' :L

n<! enQ ! P de�ned by the rule '(enr) = pnenr may be extended to a

homomorphism ' : Q ! P . Hence there exists p 2 P such thatP

n<! pnenQ �

pQ = Im'. Since pQ is a �nitely generated R-module, there is k < ! for which

pQ � Pk, soP

n<! pnenQ � Pk, a contradiction. �

Proposition 3.7. If R is a right non-singular right strongly steady ring then its

maximal ring of quotients Qmax is semi-simple.

Proof. First, note that Qmax is a �nitely generated R-module by Corollary 3.4 and

Qmax is a regular and self-injective ring by [16, Proposition XII.2.2 and Corol-

lary XII.2.3]. Therefore, from [12, Theorem 10.22], Qmax is a direct product of

rings Qmax = Q1 �Q2 �Q3 such that Q1 is of type If , Q2 is of type IIf , and Q3

is a purely in�nite ring. We view all these factors of Qmax as two-sided ideals of

Qmax.

Suppose Q2 6= 0. Then for every maximal two-sided ideal M which contains

Q1 and Q3 we deduce, using [12, Theorem 10.29], that Qmax=M is of type IIf . It

follows that Qmax=M is not artinian by [12, Theorem 10.26]. Then R\M is a two-

sided ideal of R and R=R\M can be embedded into Qmax=M such that Qmax=M is

�nitely generated as a right R=R\M -module. Then R=R\M is not right steady as

a consequence of Lemma 3.6 (2), a contradiction, hence Qmax = Q1�Q3. Moreover,

Qmax has no purely in�nite factors. Note that every injective module over a purely

in�nite ring is !-reducing by [12, Proposition 5.8] and [17, Example 2.8], so any non-

�nitely generated injective module over Q3 is non-�nitely generated !-reducing as

an R-module by Lemma 3.6(1). Hence by Lemma 2.2 Qmax is directly �nite of type

If . Then Qmax�=Qn2I Mn(Sn) where Sn, n < ! are abelian regular rings and I is

a countable set by [12, Theorem 10.24]. Assume I is in�nite. Then Qmax contains

a non-�nitely generated self-small right ideal, which is torsion-free as an R-module,


so it is a non-�nitely generated self-small R-module by Lemma 3.3(2). Then I

is a �nite set. Using again Lemma 3.3(2) we deduce that for every n 2 I every

small ideal of Mn(Sn) is �nitely generated. Then every small ideal of Sn is �nitely

generated since Sn and Mn(Sn) are Morita equivalent rings. As a consequence of

the proof of [9, Theorem 2.8] we deduce that Sn are semi-simple rings for all n 2 I

and the proof is complete. �

From this result we obtain a �rst characterization theorem.

Theorem 3.8. The following conditions are equivalent for a non-singular commu-

tative ring R:

a) R is artinian;

b) R is semi-simple;

c) R is (right) strongly steady.

Proof. a) ) b) It su�ces to prove that J = 0. If s 2 J \ Soc(R), then Soc(R) �

Ann(s). As R is non-singular, s = 0, hence J = 0. The implication b) ) a) is

obvious.

b) ) c) By Proposition 2.5.

c) ) b) By Proposition 3.7 and [13, Corollary 3.97]. �

In fact, the conclusion \R is artinian" is valid for all right non-singular rings.

Theorem 3.9. Let R be a right non-singular right strongly steady ring. Then R

is right artinian.

Proof. Applying the Proposition 3.7 and [16, Theorem XII 2.5] we deduce that

R is of �nite rank. Therefore (Qmax)R �= E(R) �=L

i�mE(Si) for uniform R-

submodules (right ideals) Si of R, i � m. Here E(Si) denotes the injective envelope

of Si. Note that every E(Si) is simple as a right Qmax-module.

We show that every non-zero R-submodule N of E(Si) is self-small, so �nitely

generated. Let N =Sn<! Ni for an strictly increasing chain of non-zero submod-

ules. Since Si is essential in E(Si), Si \Nn 6= 0 for all n. For an arbitrarily n < !,

take ' 2 EndR(N) such that '(Nn) = 0. Then ' can be extended to a homo-

morphism ' 2 EndR(E(Si)). Since E(Si) is a torsion-free module (concerning

maximum Gabriel topology) and E(Si) �= Si Qmax, we have a natural bijection

EndR(E(Si))! EndQmax(E(Si)) by Lemma 3.2. Hence ' is aQmax-endomorphism

of a simple Qmax-module which is not mono, so ' = 0. Applying Proposition 2.7,

we obtain that N is a self-small module. Then E(Si) are noetherian R-modules for

all i � m, hence Qmax is a noetherian R-module since it is a �nite direct sum of

noetherian modules.

Therefore R is a right noetherian ring. Since R is a right strongly steady ring,

Qmax�= E(R) is �nitely generated as a right R-module. Then R is right artinian

as a consequence of [4, Corollary 5.1]. �


Corollary 3.10. Let R be a right nonsingular ring which is strongly steady. Then

its maximal ring of right quotients is artinian and noetherian as a right R-module.

The conditions R is right artinian does not imply R is right strongly steady.

Example 3.11. The ring R =

�Q R0 R

�is non-singular right artinian but it is not

right strongly steady.

Proof. It is not hard to see that the maximal right ring of quotients of R is Qmax =�R RR R

�which is self small as an R-module by Corollary 3.4. Since Qmax is an

in�nitely generated R-module, R is not right strongly steady. �

Theorem 3.12. The following conditions are equivalent for a non-singular semi-

prime ring R:

a) R is semi-simple;

b) R is right strongly steady.

Proof. a))b) By Proposition 2.5.

b))a) We apply Proposition 3.7 to obtain that the maximal ring of quotient is

semi-simple. Applying [16, Proposition XV.3.3], Qmax is the classical ring of right

quotients of R. Using [14, Exercise II.16] we deduce that R and its classical ring of

quotients coincide and the proof is complete. �

The following example shows that the hypothesis \R is semi-prime" is not su-

per uous. Moreover, it follows that another important class of (non-singular) rings

is included in the class of strongly steady rings.

Example 3.13. Every upper triangular matrix ring Tn(K) over a division ring K

is strongly steady.

Proof. The proof is by induction on n. If n = 1 the property is obvious. Supposing

that Tn(K) is a strongly steady ring, we will prove that Tn+1(K) is strongly steady.

We observe that the right socle of Tn+1(K) is the ideal S which consists in all ma-

trices with 0 on the �rst n columns. Moreover S is the smallest essential ideal

of Tn+1(K), hence every singular Tn+1(K)-module is annihilated by S. There-

fore a singular Tn+1(K)-module M is a Tn+1(K)=S-module and M is self-small

as a Tn+1(K)-module if and only if it is self-small as a Tn+1(K)=S-module. Since

Tn+1(K)=S �= Tn(K), a singular self-small Tn+1(K)-module is �nitely generated by

the hypothesis. Moreover, every nonsingular right Tn+1(K)-module is projective

(see the proof of [8, 12.21]), hence it is a direct sum of projective cyclic ideals since

Tn+1(K) is an exchange ring, and it follows that a nonsingular self-small Tn+1(K)-

module is �nitely generated. To close the proof, it is enough to observe that every

Tn+1(K)-module is a direct sum of a singular module and a non-singular module

by [16, Section VI.8]. �

Using Lemma 2.4 and [8, 12.21], we obtain


Corollary 3.14. If R is a non-singular and �-extending ring then every self-small

R-module is �nitely generated.

Using Corollary 3.4 we can deduce directly that for a right nonsingular ring

which is right strongly steady ring the right R-module Qmax is �nitely generated.

In the following example we show that the condition \Qmax is �nitely generated"

does not imply that the right nonsingular ring is strongly steady.

Example 3.15. There exists a right nonsingular ring R with simple maximal ring

of quotients Qmax such that Qmax is �nitely generated as a right R-module and R

is not strongly steady. (There exists a right nonsingular ring R which is not right

artinian with simple maximal ring of quotients Qmax such that Qmax is �nitely

generated as a right R-module).

Proof. We consider the ring R =

0@ R R R

0 Q R0 0 R

1A. It is not hard to see that R is

not right artinian. If we calculate a cyclic right ideal generated by an element of

R we observe that it contains one of the ideals which have only 0 on the �rst two

columns and two 0 and one R on the last column. Therefore, the right socle of R

is S = Soc(RR) =

0@ 0 0 R

0 0 R0 0 R

1A.

Using [13, Lemma 7.2] we deduce that R is right nonsingular. Moreover, by

direct calculations it follows that S is a dense ideal of R and it is minimal (since

every proper R-submodule of S is not essential in R, [13, Examples 8.3]). Hence

by [13, Theorem 13.22] we deduce Qmax =Mat3�3(R).

The right R-module Qmax(R) is generated by E11, E12 and E13,where Eij de-

notes the matrix which have 1 on the position (i; j) and 0 on the other positions. �

Now we can complete Theorem 3.9.

Proposition 3.16. Every right noetherian right strongly steady ring is right ar-

tinian.

Proof. As R=N(R) contains no nilpotent ideal, R=N(R) is a right non-singular

strongly steady ring by [16, Lemma II.2.5], and it is semi-prime. Therefore it

is semi-simple by Theorem 3.12. By [16, Corollary XV.1.3] N(R) � J(R), and

J(R)=N(R) � J(R=N(R)) = 0 from which it follows N(R) = J(R). Applying

[16, Lemma XV.1.4] we deduce that J(R) is nilpotent, which implies that R is a

right artinian ring as a consequence of Hopkins-Levitzki Theorem [14, Theorem

4.15]. �

Theorem 3.17. For a commutative noetherian ring R, the following properties are

equivalent:

a) R is artinian;

b) R is (right) strongly steady.


Proof. The implication b))a) follows by Proposition 3.16 and the reverse one holds

true by Proposition 2.9. �

References

[1] U. Albrecht Quasi-decompositions of abelian groups and Baer's Lemma, Rocky Mount. J.Math., 22 (1992), 1227{1241.

[2] D.M. Arnold, C.E. Murley Abelian groups, A, such that Hom(A;�) preserves direct sums ofcopies of A., Pac. J. Math. 56 (1975), 7{20.

[3] J.E. Bj�ork Rings satisfying certain chain conditions, J. Reine Angew. Math. 245 (1970),63{73.

[4] J.E. Bj�ork Conditions which imply that subrings of artinian rings are artinian J. ReineAngew. Math. 247 (1972), 123{138.

[5] S. Breaz The quasi-Baer-splitting property for mixed abelian groups, J. Pure Appl. Algebra,191 (2004), 75{87.

[6] R. Colpi, C. Menini On the structure of ?-modules, J. Algebra 158 (1993), 400{419.[7] R. Colpi and J. Trlifaj Classes of generalized ?-modules, Comm. Algebra 22 (1994), 3985{

3995.[8] N.V. Dung, D. Van Huynh, P.F. Smith, R. Wisbauer Extending modules Pitman Research

Notes in Mathematics Series, 313., New York, 1994.[9] P.C. Eklof, K.R. Goodearl and J. Trlifaj Dually slender modules and steady rings, Forum

Math. 9 (1997), 61{74.[10] L. Fuchs In�nite Abelian Groups II, Academic Press, 1973.[11] J. L. G�omez Pardo, G. Militaru, C. N�ast�asescu When is HOM(M;�) equal to Hom(M;�)

in the cahegory R� gr?, Comm. Algebra, 22 (1994), 3171{3181.[12] K. R. Goodearl Von Neumann Regular Rings, London 1979, Pitman, Second Ed. Melbourne,

FL 1991, Krieger.[13] T.Y. Lam Lectures on Modules and Rings, Springer-Verlag, New York, 1999.[14] T.Y. Lam A First Course in Noncommutative rings, Springer-Verlag, New York, 1991.[15] A. Orsatti, N. Rodin�o On the endomorphism ring of an in�nite-dimensional vector space,

Abelian groups and modules (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dor-drecht, 1995, 395{417.

[16] B. Stenstr�om Rings of quotients Die Grundlehren der Mathematischen Wissenschaften, Band217, Springer-Verlag, New York-Heidelberg, 1975.

[17] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents. Abelian groups andmodules (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995, 467{473.

[18] J. Trlifaj Every ?-module is �nitely generated, J. Algebra 169 (1994), 392{398.[19] J. Trlifaj Almost ?-modules need not be �nitely generated, Comm. Algebra 21 (1993), 2453{

2462.[20] J. �Zemli�cka Steadiness is tested by a single module, Contemporary Mathematics, 273 (2001),

301{308.[21] J. �Zemli�cka !1-generated uniserial modules over chain rings, Comment. Math. Univ. Caroli-

nae 45 (2004), 403{415.[22] J. �Zemli�cka Steadiness of regular semiartinian rings with primitive factors artinian,

J.Algebra, 304 (2006), 500{509.

"Babes�-Bolyai" University, Faculty of Mathematics and Computer Science, Str. Mi-

hail Kog�alniceanu 1, 400084 Cluj-Napoca, Romania

E-mail address: [email protected]

Department of Algebra, Charles University in Prague, Faculty of Mathematics and

Physics Sokolovsk�a 83, 186 75 Praha 8, Czech Republic


B. WHEN PRODUCTS OF SELF-SMALL MODULES ARE

SELF-SMALL

JAN �ZEMLI�CKA

Abstract. A module M is called self-small if the functor Hom(M;�) com-mutes with direct sums of copies of M . The main goal of the present paper isto construct a non-self-small product of self-small modules without non-zerohomomorphisms between distinct ones and to correct an error in a claim aboutproducts of self-small modules published by Arnold and Murley in a funda-mental paper on this topic. The second part of the paper is devoted to thestudy of endomorphism rings of self-small modules.

The notion of a compact object of a category, i.e. an object c for which the

covariant functor Hom(c;�) commutes with all direct sums, has appeared as a

natural tool in many branches of module theory. Smallmodules, which are precisely

compact objects of the category of all modules over a ring, have been useful in

the study of the structure theory of graded rings [8] and almost free modules [5].

The most recent motivation of this topic comes from the context of representable

equivalences of module categories [3, 4]. The central notion of the present paper,

a self-small module, which can be de�ned as a compact object c of the category of

direct summands of all direct sums of copies of c, was introduced in [1] as a tool for

generalization of Baer's lemma [6, 86.5]. Nevertheless, self-small modules, similarly

to small modules, turn out to be important in the study of generalization of Morita

equivalence [3, 4].

The �rst two sections of the work [1] are devoted to describe basic properties of

general self-small modules. Among correct and classical structure results about self-

small modules there is one [1, Corollary 1.3] which has a too weak hypothesis (and

a too quick proof) as we show in Proposition 1.4 and Example 1.5 in the present

paper. The construction of Proposition 1.4 is a consequence of a more general

fact that a product of a representable set of all simple modules over a non-steady

abelian regular ring is not a self-small module, but Hom(S; T ) = 0 for an arbitrary

pair of non-isomorphic simple modules S and T (Corollary 1.3). A correct version

of [1, Corollary 1.3] is proved in Proposition 1.6 and in the rest of the section we

investigate several classes of examples for which the hypothesis of Proposition 1.6

is satis�ed, i.e. when the conclusion of [1, Corollary 1.3] holds true. The second

section of this paper illustrates the limitations of using endomorphism rings to

2000 Mathematics Subject Classi�cation. 16D10 (16S50, 16D70).Key words and phrases. Self-small module, strongly steady ring, endomorphism ring.This work is part of the research project MSM 0021620839, �nanced by M�SMT.

36

B. WHEN PRODUCTS OF SELF-SMALL MODULES ARE SELF-SMALL 37

detect whether a module is self-small. Theorems 2.5, and 2.9 respectively show

that over a non-trivial commutative principal ideal domain with zero Jacobson

radical, and over a non-artinian abelian regular rings respectively there exists a

pair of a self-small module and a non-self-small module with the same isomorphism

rings.

We consider rings as unitary and associative, and a module means a right R-

module over an arbitrary ring R. Let M be a module and N its submodule, then

de�ne VM (N) = ff 2 End(M)j f(N) = 0g. Recall thatM is a self-small module i�

for each increasing chain of submodules Nn � Nn+1 of M for which M =Sn<! Nn

there exists n such that VM (Nn) = 0 [1, Proposition 1.1 (b)]. Moreover, a small

module M can be characterized in similar way: if M =Sn<! Nn for an increasing

chain of submodules of N , then there exists n such that Nn = M . Note that

a �nitely generated module is a compact object of any full subcategory of the

category of all modules. Thus the class of all self-small modules contains the class

of all small modules and it contains all �nitely generated modules. Both inclusions

are strict in general, as it is shown in [1, 10], which leads to the de�nitions of a right

strongly steady (right steady) ring as a ring over which every right self-small (small)

module is necessary �nitely generated. Several necessary and su�cient conditions

of strongly steady rings are proved in [2], properties of steady rings are investigated

in [5, 9, 10, 11].

Let M be a module. Jacobson radical of M is denoted by J(M), and for � 2M

or � = M , the annihilator of � in a ring R is denoted by AnnR(�). Recall that an

abelian regular ring has every principal right and left ideal generated by a central

idempotent. For basic properties of abelian regular rings we refer to [7, Chapter

III].

1. Products

Denote by SR a representative set of simple modules over a ring R. Before

we start to construct an example of a non-self-small product of self-small modules

without non-zero homomorphisms between distinct modules, we prove the following

criterion.

Theorem 1.1. Let R be an abelian regular ring and M a module containing a copy

of every simple module as a submodule. Then M is self-small i� it is small.

Proof. First, note that every small module is self-small in general, so it is enough to

prove the direct implication. Suppose thatM is not small and letM =SnMn for a

strictly increasing chain of submodules Mn (Mn+1 �M . As every simple module

over an abelian regular ring is injective by [7, Proposition 6.18], every R-module

has zero Jacobson radical. Thus M=Mn has a simple quotient for each n. Since M

contains a copy of every simple submodule, there exists a non-zero homomorphism

38 JAN �ZEMLI�CKA

g : M=Mn ! M . Finally, put fn = g�n where �n is the natural projection of M

onto M=Mn. For every n we have found a non-zero endomorphism fn such that

fn(Mn) = 0, i.e. M is not self-small by [1, Proposition 1.1 (b)]). �

AsL

S2SRS �QS2SR

S, we may formulate the following consequences:

Corollary 1.2. Let R be an abelian regular ring. ThenQS2SR

S is a self-small

module i� it is small.

Corollary 1.3. Let R be an abelian regular ring which is right steady and such

thatQS2SR

S is in�nitely generated. ThenQS2SR

S is not self-small.

Proposition 1.4. Let R be a countable non-artinian abelian regular ring. ThenQS2SR

S is not self-small.

Proof. Since SR is in�nite, the moduleQS2SR

S is not countable, hence it is in-

�nitely generated. As R is right steady by [11, Theorem 9] we may apply Corol-

lary 1.3. �

Example 1.5. Let R be the subring of the ring Q! generated by Q(!) and by all

constant functions cq 2 Q! (i.e. R is formed by the eventually constant functions,

cf. the constructions in [5] and [9]). Note that R is a countable abelian regular

ring containing the in�nitely generated ideal Q(!), henceQS2SR

S is not self-small

by Proposition 1.4. Since every simple module is self-small and HomR(S; T ) = 0

for every di�erent modules S; T 2 SR, we have constructed a counterexample to [1,

Corollary 1.3].

[1, Corollary 1.3] can be corrected in the following way:

Proposition 1.6. Let (Mij i 2 I) be a system of self-small modules satisfying

the condition HomR(Qj2InfigMj ;Mi) = 0 for each i 2 I. Then

Qj2I Mj is a

self-small module.

Proof. Put M =Qj2I Mj and suppose M =

Sn<! Nn for a chain of submodules

Nn � Nn+1. Denote by �i and �i respectively the natural projection of M onto Mi

and the natural injection of Mi into M respectively.

First, note that there exists a minimal ni such that VMi(�i(Nni)) = 0 for ev-

ery i 2 I. If fnij i 2 Ig is not bounded, �x for every n < ! some in so that

VMin(�in(Nn)) 6= 0, i.e. �in(Nn) 6= Min and we can take m 2 M for which

�in(m) =2 �in(Nn). Hence m =2 Nn for every n, a contradiction (this part of the

proof works as in the proof of [1, Corollary 1.3]).

Thus fnig is bounded, so there exists n such that VMi(�i(Nn)) = 0 for each i 2 I.

Note that m� �i�i(m) 2Qj2InfigMj , where we consider

Qj2KMj in natural way

as a submodule ofQj2I Mj whenever K � I. Hence �i'(m � �i�i(m)) = 0 for

every i 2 I, m 2 M and ' 2 End(M) by the premise. Let ' 2 End(M) such

that '(Nn) = 0. Then 0 = �i'(p) = �i'(�i�i(p)) for arbitrary p 2 Nn. Since


VMi(�i(Nn)) = 0 and �i'�i 2 End(Mi) we get �i'(m) = �i'(�i�i(m)) = 0 for

every m 2M and i, which implies that ' = 0 and VM (Nn) = 0. �

Applying Proposition 1.6 we will show that we can produce new examples of

self-small modules as products of self-small modules (which was an original aim of

[1, Corollary 1.3]).

Lemma 1.7. Let R be a commutative principal ideal ring and let T 2 SR. Then

HomR(Y

S2SRnfTg

S; T ) = 0:

Proof. Put iR = AnnR(T ) for a suitable i 2 R. Fix an arbitrary simple module

S non-isomorphic to T . Since iR * AnnR(S), Si 6= 0 and so Si = S. Hence

(QS2SRnfTg

S)i =QS2SRnfTg

S. If we take a homomorphism ' :QS2SRnfTg

S !

T , it holds true thatQS2SRnfTg

S = (QS2SRnfTg

S)AnnR(T ) � ker ', i.e. ' =

0. �

Corollary 1.8. If R is a commutative principal ideal ring,QS2SR

S is a self-small

module.

As it is shown in the sequel, the stronger version of the hypothesis of [1, Corol-

lary 1.3] can be easily veri�ed for direct summands of rings. First, we make an

observation which follows from the fact that HomR(M; eR) = 0 for every central

idempotent e contained in the annihilator of M :

Lemma 1.9. Let feij i 2 Ig be an orthogonal set of central idempotents of a ring

R. Then HomR(enR; eiR) = 0 and HomR(Qj 6=i ejR; eiR) = 0 for every i 6= n.

Proposition 1.10.Qi2I eiR is a self-small module for a ring R and every orthog-

onal set of non-zero central idempotents feij i 2 Ig.

Proof. Since HomR(Qj 6=i ejR; eiR) = 0 by Lemma 1.9 and every eiR is self-small,

we may apply Proposition 1.6. �

We �nish this section with a characterization of strongly steady abelian regular

rings.

Proposition 1.11. Over every non-artinian abelian regular ring there exists an

in�nitely generated self-small module.

Proof. As R is a non-artinian abelian regular ring, it contains an in�nite orthogonal

set of non-zero central idempotents, feij i < !g. IfQi<! eiR is an in�nitely

generated module, it is an example of an in�nitely generated self-small module by

Proposition 1.10.

Now, suppose that S =Qi<! eiR is �nitely generated. As

Qi2K eiR is a direct

summand, it is a �nitely generated module as well for every K � !. Let F is a

�lter on ! and de�ne '(F) = fs 2 Sj9X 2 F : �i(s) = 0 8i 2 Xg, a submodule of

S, where �i is the natural projection onto eiR. If F is a non-principal ultra�lter,


then the same argument as in [10, Lemma 2.4 (ii)] shows that '(F) is a small, so

self-small module. Moreover, '(F) is an in�nitely generated R-module since it is

an in�nitely generated S-module by [10, Lemma 2.2 (ii)] where S =Qi<! eiR is

considered as a ring. We have found an in�nitely generated self-small module. �

By [2, Proposition 2.5 (ii)] and since every artinian regular ring is semisimple,

we obtain:

Corollary 1.12. An abelian regular ring is strongly steady i� it is semisimple.

2. Endomorphism-rings

As it is shown already in [1], an endomorphism ring of a module can serve as a

useful tool when we ask whether the module is self-small. So for example M is a

self-small module if End(M) is countable [1, Corollary 1.4]. We show in this section

that the structure of an endomorphism ring of a module need not allow to decide

whether the module is self-small over a general ring. Nevertheless, note �rst that

the self-smallness of a module over a commutative artinian ring may be recognized

in the structure of its endomorphism ring.

Proposition 2.1. Let R be a commutative artinian ring and let M be a module.

Then M is self-small, i� EndR(M) is a left artinian ring.

Proof. Let M be a self-small module and put E = EndR(M). Then M is �nitely

generated by [2, Proposition 2.9], i.e. M =P

i�nmiR. Since we may suppose

that M is faithful over R and identify elements of R with endomorphisms, we may

consider the ring R as a subring of (the center of) E. As M =P

i�nmiR =Pi�nEmi is an artinian R-module, it has the structure of an artinian left E-

module. Hence every left cyclic E-module Emi�=E E=AnnE(mi) is artinian as

well. Since R is commutative,Ti�nAnnE(mi) = f' 2 Ej '(mi) = 0 8i � ng =

f' 2 Ej '(M) = 0g = 0. It shows the natural E-homomorphism of E intoQi�nE=AnnE(mi) is a monomorphism, hence E is a left artinian ring.

The reverse implication follows immediately from [1, Proposition 1.1.(d)]. �

Now, we �nd a set of modules such that the endomorphism ring of the direct

sum is isomorphic to the endomorphism ring of the product over this set.

Lemma 2.2. Let (Mij i 2 I) be a system of modules such that HomR(Mj ;Mi) = 0

whenever i 6= j. Then EndR(L

i2I Mi) �=Qi2I EndR(Mi). Moreover, if HomR(

Qj2InfigMj ;Mi) =

0 for each i 2 I, then EndR(Qi2I Mi) �=

Qi2I EndR(Mi).

Proof. De�ne mappings 1 : EndR(L

i2I Mi)!Qi2I End(Mi) and 2 : EndR(

Qi2I Mi)!Q

i2I End(Mi)) by the formula k(') = (�i'�ij i 2 I), k 2 f1; 2g, where �i is

the natural projection onto Mi and �i denotes the natural injection of Mi. Ob-

viously, k is in the both cases a surjective Z-homomorphism and k(1) = 1.


Since '�i(Mi) � Mi (�L

i2I Mi or �Qi2I Mi) by the hypothesis, we ob-

tain that �i�i'�i = '�i, hence [k('1'2)](i) = �i'1'2�i = (�i'1�i)(�i'2�i) =

[k('1)k('2)](i) for every i 2 I, k 2 f1; 2g. We have proved 1 and 2 surjec-

tive ring homomorphisms.

It is obvious that 1 is injective and so it is an isomorphism. To conclude that

2 is an isomorphism it remains to prove that ker 2 = 0. Fix any non-zero

' 2 EndR(Qi2I Mi). Then there exists i 2 I such that �i' 6= 0. Now, it follows

by the hypothesis thatQj2InfigMi � ker �i', hence [2(')](i) = �i'�i 6= 0. �

Corollary 2.3. EndR(L

S2SRS) �=

QS2SR

EndR(S), for an arbitrary ring R.

Proposition 2.4. Let R be a commutative non-artinian principal ideal domain

with J(R) = 0. Then SR is in�nite and EndR(QS2SR

S) �=QS2SR

EndR(S).

Proof. Note that SR is in�nite since R contains in�nitely many maximal ideals

(otherwise R would be artinian). Moreover, the hypothesis of Lemma 2.2 is satis�ed

for a system of all simple modules SR by Lemma 1.7. �

Theorem 2.5. Let R be a commutative non-artinian principal ideal domain with

J(R) = 0. Then M =QS2SR

S is a self-small module, N =L

S2SRS is not

self-small and EndR(M) �= EndR(N).

Proof. M is self-small by Corollary 1.8 and N is not self-small clearly. Using

Proposition 2.4 and Corollary 2.3 we get the ring isomorphisms EndR(QS2SR

S) �=QS2SR

EndR(S) �= EndR(L

S2SRS). �

Corollary 2.6. Let R be a commutative ring such that J(R) is not maximal and

R=J(R) is principal ideal domain. Then there exists a pair of a self-small module

M and a non-self-small module N such that EndR(M) �= EndR(N).

Example 2.7. Denote by P the set of all prime numbers. By Theorem 2.5

M =Qp2P Zp is a self-small abelian group, N =

Lp2P Zp is not self-small and

EndR(M) �= EndR(N).

Another class of examples can be constructed over rings containing an in�nite

set of central orthogonal idempotents.

Proposition 2.8. Let R be a ring and feij i < !g an orthogonal set of non-zero

central idempotents. Then EndR(Qi<! eiR)

�=Qi<! eiR.

Proof. Since HomR(Qj 6=i ejR; eiR) = 0 for every i by Lemma 1.9, we may apply

Lemma 2.2 on the system of modules feiRj i < !g. �

Theorem 2.9. Let R be a non-artinian abelian regular ring. Then there exists a

pair of a self-small moduleM and a non-self-small module N such that EndR(M) �=

EndR(N).


Proof. Since R is not artinian, it contains an in�nite set of orthogonal idempotents

which are obviously central, take such a set feij i < !g and putM =Qi<! eiR and

N =L

i<! eiR. Note that M is self-small by Proposition 1.10, and it is obvious

that N is not self-small. Finally, applying Lemma 2.2 and Proposition 2.8 we get

that EndR(M) �=Qi<! eiR

�= EndR(N). �

References

[1] Arnold, D.M. , Murley, C.E. Abelian groups, A, such that Hom(A,-) preserves direct sums ofcopies of A. Pac. J. Math. 56 (1975): 7{20.[2] Breaz, S., �Zemli�cka, J. When every self-small module is �nitely generated. J. Algebra (2007),doi:10.1016/j.jalgebra.2007.01.037.[3] Colpi, R., Menini, C. On the structure of ?-modules. J. Algebra 158 (1993): 400{419.[4] Colpi, R., Trlifaj, J. Classes of generalized ?-modules. Comm. Algebra 22 (1994): 3985{3995.[5] Eklof, P.C., Goodearl, K.R., Trlifaj, J. Dually slender modules and steady rings. Forum Math.

9 (1997): 61{74.[6] Fuchs, L. In�nite abelian groups II. Academic Press: New York-London, 1973.[7] Goodearl, K. R. Von Neumann regular rings. Pitman: London 1979, Second Ed. Krieger:Melbourne, FL 1991.[8] G�omez Pardo, J. L., Militaru, G., N�ast�asescu, C. When is HOM(M;�) equal to Hom(M;�)in the category R� gr? Comm. Algebra 22 (1994): 3171{3181.[9] R�u�zi�cka, P., Trlifaj, J., �Zemli�cka, J. Criteria of steadiness. In Abelian Groups, Module Theory,

and Topology. Marcel Dekker: New York 1998, pp. 359{372.[10] Trlifaj, J. Steady rings may contain large sets of orthogonal idempotents. Proc. Conf. \AbelianGroups and Modules", Padova, Italy, 1994; Kluwer: Boston, 1995, pp. 467-473.[11] �Zemli�cka, J., Trlifaj, J. Steady ideals and rings. Rend. Sem. Mat. Univ. Padova 98 (1997):161{172.

Katedra algebry MFF UK, Sokolovsk�a 83, 186 75 Praha 8, Czech Republic


Chapter 3

Small modules and steady rings

The chapter presents two particular solutions of general problem on ring-theoreticaldescription of steady rings. The �rst text provides instead a ring-theoreticalproperty an idea of a test module of steadiness and the second one gives a ring-theoretical characterization in the particular class of regular semiartinian ringswith primitive factors artinian:

C. Jan �Zemli�cka, Steadiness is tested by a single module, in: Kelarev, A. V.(ed.) et al., Abelian groups, rings and modules. Proceedings of the AGRAM2000 conference, Perth, Australia, July 9-15, 2000. American MathematicalSociety (AMS), Providence, RI. (2001), Contemp. Math. 273, 301{308.

D. Jan �Zemli�cka, Steadiness of regular semiartinian rings with primitive factors

artinian, J. Algebra 304/1 (2006), 500{509.

43

C. STEADINESS IS TESTED BY A SINGLE MODULE

JAN �ZEMLI�CKA

DEDICATED TO PROFESSORS L �ASZL �O FUCHS AND LADISLAV PROCH�AZKA IN

HONOUR OF THEIR 75-TH AND 70-TH BIRTHDAY

Abstract. The main goal of the paper is to provide a module-theoretic cri-terion of the steadiness of a ring. We show that the existence of an in�nitelygenerated dually slender module depends on the existence of an in�nitely gen-erated dually slender submodule of a single module. In particular, steadinessis tested over a commutative regular ring R by the module R�:

Dually slender modules are the modules M for which the covariant functor

Hom(M;�) commutes with direct sums. In particular, �nitely generated mod-

ules are dually slender. Countably generated modules are not dually slender. The

notion of dually slender module dualizes the well known notions of slender modules

and slim modules [EM]. Bass noticed that a module is dually slender if and only if

it is not the union of a countable strictly increasing chain of submodules [B]. From

this characterization it simply follows that the class of all dually slender modules is

closed under homomorphic images. For some types of rings dually slender modules

form a class much larger than the �nitely generated ones. For instance, it is proved

in [CT] that the class of all dually slender modules over the endomorphism ring of

an in�nitely generated free module contains the class of all injective modules.

Rings for which the dually slender modules coincide with the �nitely generated

modules are called right steady rings. Rings satisfying various �niteness conditions

are known to be steady. For instance, right noetherian [R, CT], right perfect [CT],

and semiartinian rings of countable socle length [T2] belong to the class of right

steady rings. It is proved in [RTZ] that commutative semiartinian rings over which

no cyclic module contains an in�nitely generated dually slender submodule are

steady as well.

Although a general ring-theoretic characterization of steady rings has not been

found as yet, and so it is still an open problem, we present a characterization

that uses a single module. Assume that R is a ring. It is shown here that a ring

is right steady if and only if a single module of cardinality bounded by 22card(R)

contains no in�nitely generated dually slender submodule. It is also proved that

any representative class of dually slender modules over a commutative regular ring

is a set, and we give an estimate of the cardinality of each dually slender module.

Research supported by grants GAUK 254/2000/B MAT/MFF, GA�CR 201/00/P071 and MSM113200007.

44

C. STEADINESS IS TESTED BY A SINGLE MODULE 45

Furthermore, we present a more precise characterization of steady commutative

regular rings. A commutative regular ring R is steady if and only if R� contains no

in�nitely generated dually slender submodule.

In the rest of the paper module means a right R-module. An ideal is a two-sided

ideal. A regular ring means a von Neumann regular ring. A regular ring is called

abelian regular provided all of its idempotents are central. A ring is said to be

semilocal if its Jacobson radical is the intersection of �nitely many maximal right

ideals.

A representative set of simple modules over a ring will be denoted by Simp. R�

will denote the right R-module R� = (RR)� = HomZ(RR;Q=Z):

Let M be a module and I be a right ideal. The set of all maximal submodules of

M will be denoted by Max(M): J(M) will denote the Jacobson radical of M and

E(M) the injective envelope of M . The minimal cardinality of sets of generators of

M will be denoted by gen(M). Finally, AnnI(M) will stand for the right annihilator

of M in I, i.e. AnnI(M) = fi 2 I; Mi = 0g.

For further notation we refer to [AF] and [G].

1. General case

De�nition 1.1. Let R be a ring and � = card(R)+. De�ne the modules

T1 =Y

S2Simp

S� and T2 =Y

S2Simp

S(!):

Finite rings are noetherian, so they are right and left steady. Note that every

dually slender module over a skew-�eld is a �nitely generated semisimple module.

Lemma 1.2. Let R be a ring and M be an in�nitely generated dually slender

module such that J(M) = 0. Then there exists a homomorphism � : M ! T1 such

that �(M) is in�nitely generated. Moreover, if the ring R is commutative, then M

is embeddable in T2.

Proof. The ring R is not right steady, so it is in�nite.

Let �N denote the natural projection M !M=N for each N 2Max(M). Since

J(M) = 0, the product h : M !QN2Max(M)M=N of the homomorphisms �N

is a monomorphism. De�ne a set X � M by letting X be an arbitrary subset

of M of cardinality � if card(M) � �, and X = M if card(M) < � (recall that

� = card(R)+). Clearly, the cardinality of the module XR (�M) is at most �:

For each a 2 XR, a 6= 0, �x a module Na 2Max(M) for which a 2M nNa: Put

U = fNa; a 2 XR; a 6= 0g �Max(M): Obviously, card(U) � card(XR) � �: Let

� be the natural projection ofQN2Max(M)M=N onto

QN2U M=N . Then �hd(XR)

is a monomorphism. If X = M , �h is a monomorphism as well. If card(X) = �,

then card(�h(M)) � card(�h(XR)) � � > card(R(!)) = card(R), because R is an

in�nite ring. Hence �h(M) is an in�nitely generated module. As card(U) � � and

�h(M) is a submodule ofQN2U M=N , �h(M) is embeddable in T1.


Assume that R is a commutative ring. Then for every N 2 Max(M) there

exists an I 2 Max(R) such that MI � N . Since M is a dually slender R-module,

M=MI is a dually slender R=I-module, and so M=MI �= (R=I)(nI) for each I 2

Max(R) and for a suitable �nite number nI . AsTI2Max(R)MI = 0, the product

h : M !QI2Max(R)(R=I)

(nI) of the projections �I : M !M=MI, I 2Max(R), is

a monomorphism. Obviously, there is a natural injection i :QI2Max(R)(R=I)

(nI) !

T2, so ih embeds M into T2. �

Note that in the commutative case all �nitely generated modules with zero Ja-

cobson radical are embeddable in T2; the same holds for in�nitely generated dually

slender modules with zero Jacobson radical.

Lemma 1.3. Let R be a ring and M be a non-zero module such that J(M) = M .

Then there exists a homomorphism � : M ! E(S) for a simple module S such that

�(M) is in�nitely generated .

Proof. Fix an arbitrary non-zero element m 2 M: Then there exists a non-zero

homomorphism �� : mR ! E(S) for a suitable S 2 Simp(R): As E(S) is an in-

jective module, we can extend �� to a homomorphism � : M ! E(S): Since both

M and �(M) are non-zero and have no maximal submodule, �(M) is not �nitely

generated. �

Theorem 1.4. Let R be a ring. If R is commutative, put T = T2, otherwise put

T = T1. Then the following conditions are equivalent:

(1) R is right steady.

(2) Each dually slender submodule of T and each dually slender submodule of

R� is �nitely generated.

(3) Each dually slender submodule of T and each dually slender submodule of

every E(S); S 2 Simp, is �nitely generated.

Proof. (1)!(2) Obvious.

(2)!(3) It is well known that R� is an injective cogenerator, see for instance [S,

Chapter I, Proposition 9.3]. Thus both S and E(S) are embeddable in R� for each

S 2 Simp.

(3)!(1) Let R be non-steady and M be an in�nitely generated dually slender

module. If M=J(M) is in�nitely generated , T contains an in�nitely generated

dually slender submodule by Lemma 1.2. On the other hand, assume that M =

F + J(M) for a �nitely generated module F: Then J(M=F ) = M=F . Applying

Lemma 1.3 we get a simple module S such that E(S) contains an in�nitely generated

dually slender submodule. �

Corollary 1.5. Let R be a semilocal ring. Then R is right steady if and only if

E(R=J(R)) contains no in�nitely generated dually slender submodule.


Proof. Since R is semilocal, R=J(R) is semisimple. So the module T is semisimple

as well. Hence T contains no in�nitely generated dually slender submodule and the

assertion follows immediately from Theorem 1.4. �

Proposition 1.6. Let R be a commutative regular ring. Then each dually slender

module is embeddable in T2.

Proof. Let M be a dually slender module. It is well known that every module over

a commutative abelian regular ring has zero Jacobson radical. Hence, from Lemma

1.2 it follows that M embeds in T2. �

It is an immediate consequence of the last proposition that a representative class

of dually slender modules over a commutative regular ring is a set.

Proposition 1.7. Let R be a commutative ring. If R� contains no in�nitely gener-

ated dually slender submodule, then a representative class of dually slender modules

is a set.

Proof. Suppose that a representative class of dually slender modules is proper.

Obviously, R is non-steady, so R is in�nite. Thus there exists a dually slender

module M such that gen(M) > card(T2): By Lemma 1.2 M=J(M) embeds in T2;

so there is a submodule N ofM such that gen(N) � card(T2) and N+J(M) =M:

HenceM=N is an in�nitely generated dually slender module containing no maximal

submodule. Applying Lemma 1.3 we get an in�nitely generated dually slender

submodule of R�; a contradiction. �

In [T2] it is shown that the class of all dually slender modules over the endomor-

phism ring of an in�nitely generated free module contains the class of all injective

modules. Thus any representative class of dually slender modules over this ring is

a proper class.

2. Steadiness of commutative regular rings

De�nition 2.1. Let M be a module and I be a two-sided ideal which is maximal

as a right ideal. Then de�ne

dI(M) = dimR=I((M +MI)=MI):

Clearly, the notion is well de�ned because R=I is a skew-�eld.

Note that dI(N) = dimR=I(N + MI=MI) for all modules M and N over an

abelian regular ring, N �M . Indeed, N \NI = N \MI over each abelian regular

ring, so (N +NI)=NI �= N=(N \NI) = N=(N \MI) �= (N +MI)=MI:

If M is a dually slender module, dI(M) < ! for each maximal ideal I:

Lemma 2.2. Let R be an abelian regular ring and M be a dually slender module.

(1) Assume that N is a submodule of M such that dI(N) = dI(M) for each

I 2Max(R): Then M = N:


(2) Assume that J is an ideal satisfying dI(M) = 0 for each I 2Max(R) such

that J � I: Then M =MJ:

Proof. Each module over an abelian regular ring has zero Jacobson radical. In par-

ticular, every non-zero dually slender module over an abelian regular ring contains

a maximal submodule.

(1) Let I be an arbitrary maximal ideal. Since (N + MI)=MI = M=MI,

we get following isomorphisms: 0 = M=(N +MI) �= (M=N)=((N +MI)=N) �=

(M=N)=((M=N)I): Hence dI(M=N) = 0 for each maximal ideal I, so the module

M=N contains no maximal submodule. From the hypothesis it follows that M=N

is the zero-module, and so M = N:

(2) Assume that J � I. Then (MJ +MI)=MI = 0: Therefore dI(MJ) = 0 =

dI(M):

On the other hand, assume that J 6� I: Since I is maximal, J + I = R: Thus

MJ +MI = M(J + I) = M , and so dI(MJ) = dI(M) for every maximal ideal I:

Hence (1) implies the assertion. �

It follows from Proposition 1.6 that the cardinality of each dually slender module

over an in�nite commutative regular ring is bounded by 22card(R)

: The following

assertion improves on the estimate of the cardinality.

Corollary 2.3. Let R be an in�nite commutative regular ring. Let M be a dually

slender module. Then gen(M) � card(M) � 2card(R):

Proof. SinceM is a dually slender module, dI(M) < ! for each I 2Max(R). Hence

there exist �nitely generated modules FI � M such that FI +MI = M . Thus,

by Lemma 2.2 (1),P

I2Max(R) FI = M: As the cardinality of Max(R) is bounded

by 2card(R) and the cardinality of each FI is bounded by card(R), card(M) �

2card(R): �

Lemma 2.4. Assume that M is a module and M =P

i<!Mi, where the Mi; i < !;

are submodules of M such that no factor-module of Mi contains an in�nitely gener-

ated dually slender submodule. Then no factor-module of M contains an in�nitely

generated dually slender submodule.

Proof. This is an easy generalization of [ZT, Lemma 5] where the statement is

proved for Mi = miR; i < ! and M =P

i<!miR: �

Let R be an abelian regular ring. Fix a module P and an element x 2 R.

Note that PxR = PeR = Pe for a suitable central idempotent e 2 R (moreover,

xR = eR). Thus PxR is a homomorphic image of P (a homomorphism is de�ned

as multiplication by the central element e). Consequently, PxR is dually slender ,

if P is a dually slender module.

Lemma 2.5. Let R be an abelian regular ring such that no cyclic module contains

an in�nitely generated dually slender submodule. Let N be an in�nitely generated


dually slender module. De�ne the set

S = fr 2 R; gen(NrR) < !g:

Then S is an ideal. In addition, either N=NS is an in�nitely generated module

or there exists a �nitely generated submodule F such that (N=F )S = N=F and

S=AnnS(N=F ) is in�nitely generated.

Proof. Since each factor of R contains no in�nitely generated dually slender ideal,

�nitely generated modules contain no �nitely generated dually slender module

(Lemma 2.4).

First, we will prove that S is an ideal. Since R is abelian regular, it is su�cient

to show that S is a right ideal.

Fix two elements r; s 2 S. We have observed that N(r + s)R is a homomorphic

image of N; hence N(r + s)R is a dually slender module. Moreover, N(r + s)R is

a submodule of the �nitely generated module NrR+NsR: It follows from Lemma

2.4 that N(r + s)R is �nitely generated. Thus r + s 2 S: Now �x elements i 2 S

and r 2 R: Similarly, NirR is a dually slender submodule of the �nitely generated

module NiR, hence NirR is �nitely generated as well. Therefore, ir 2 S: This

proves that S is a right ideal.

Assume that N=NS is �nitely generated. Then there exists a �nitely generated

module F � N for which F + NS = N: Thus N=F = (N=F )S: If S=AnnS(N=F )

were �nitely generated, there would exist a central idempotent e 2 S such that

S = eR+AnnS(N=F ). Obviously, Ne+F = N; so N would be �nitely generated,

a contradiction. Thus S=AnnS(N=F ) is in�nitely generated. �

Lemma 2.6. Let R be a non-steady abelian regular ring. Then:

(1) There exist an in�nitely generated dually slender module M and an ideal J

such that MJ =M and gen(J=AnnJ(M)) � !:

(2) There exist an in�nitely generated dually slender module M and a strictly

increasing chain of ideals (J�; � < �) for an uncountable cardinal � =

cf(�) such that M =S�<�MJ� and M 6=MJ� for each � < �:

Proof. (1) If there exists a factor-ring of R which contains an in�nitely generated

dually slender ideal I, put M = I and let J be a lifting of I to the ring R. Clearly,

MJ = II = I =M since R is a regular ring.

Let no factor of R contain an in�nitely generated dually slender ideal. Since R is

non-steady, there exists an in�nitely generated dually slender module N . Now ap-

plying Lemma 2.5 we get an ideal S such that eitherN=NS is an in�nitely generated

module or there exists a �nitely generated submodule F such that (N=F )S = N=F

and S=AnnS(N=F ) are in�nitely generated ideals.

Assume that N=NS is a �nitely generated module. Put M = N=F and J = S:

Obviously, MJ =M and J=AnnJ(M) is in�nitely generated.

On the other hand, let N = N=NS be an in�nitely generated module.


Assume that there exists a central idempotent e 2 RnS for which gen(Ne) < !.

>From the de�nition of S it follows that gen(Ne) � !: Moreover, there is a �nitely

generated module F � Ne for which Ne + NS = F + NS: Since R is an abelian

regular ring, we get that Ne � NSe + Fe � NeS + F: Hence (Ne=F )S = Ne=F:

If S=AnnS(Ne=F ) were �nitely generated, there would exist a central idempotent

f 2 S such that Nef+F = Ne; so Ne would be �nitely generated, a contradiction.

Now put M = Ne=F and J = S. We have proved that M =MJ , and J=AnnJ(M)

is in�nitely generated.

Finally, assume that there exists no idempotent e 2 RnS such that gen(Ne) < !.

If Nx = 0 for an element x 2 R, 0 = NxR = Ne for a central idempotent such that

xR = eR. Then e 2 S and xR � S. Hence N = N=NS is faithful over R = R=S:

Since R is also non-steady (indeed, N=NS is an in�nitely generated dually slender

module over R), it is not semisimple. Thus there exists an in�nitely generated

maximal ideal in R; let us denote it by I: Moreover, there is a �nitely generated

module K for which K+NI = N: Now both N andM = N=K are faithful over R;

because �nitely generated modules contain no in�nitely generated dually slender

module (Lemma 2.4). Let the ideal J be a lifting of the ideal I to the ring R.

Obviously, MJ =M and gen(J=AnnJ(M)) � !:

(2) Applying (1) we get an in�nitely generated dually slender module M and

an ideal J such that MJ = M and gen(J=AnnJ(M)) � !: Fix M and J for

which gen(J=AnnJ(M)) is minimal (but in�nite). W.l.o.g. we can suppose that

M is faithful (i.e. AnnJ(M) = AnnR(M) = 0). Note that MjR 6= M for each

j 2 J 6= R, since jR = eR 6= R for a suitable central idempotent 1 6= e 2 jR and

Mjr = Me 6= Me �M(1 � e) = M . Let (J�; � < �) be an arbitrary �ltration of

J by submodules such that cf(�) = � and card(J�) < card(J) for each � < �: By

the minimality of gen(J); MJ� 6= M for each � < �: Since M is a dually slender

module (i.e. M is not the union of any countable chain of submodules), � is not

countable. �

Now we are ready to characterize the steadiness of a commutative regular ring

in terms of the module R� (it is easy to generalize our result to abelian regular

rings).

Theorem 2.7. Let R be a commutative regular ring. Then R is steady if and only

if R� contains no in�nitely generated dually slender submodule.

Proof. Let R be a non-steady commutative regular ring. Applying Lemma 2.6 we

get an in�nitely generated dually slender module M and a strictly increasing chain

of ideals (J�; � < �) such that M =S�<�MJ� and M 6= MJ� for each � < �:

We will de�ne a sequence of maximal ideals (I� ; � < �) and a strictly increasing

sequence of ordinals (�� ; � < �) such that dI� (M) 6= 0, J�� I� and J��+1 6� I�

via trans�nite induction.

Fix an arbitrary maximal ideal I0 such that dI0(M) 6= 0 and let �0 = 0:


Assume that both I� and �� are de�ned. Since dI� (M) 6= 0, M 6� MI� . More-

over, M =S�<�MJ�, hence there exists � > �� such that MJ� 6� MI� . Now

put ��+1 = �. Obviously, J��+1 6� I� . Applying Lemma 2.2(2) (for J = J��+1),

we get I�+1 2Max(R) for which J��+1 � I�+1 and dI�+1(M) 6= 0:

If is a limit ordinal, put � = sup(�� ; � < ) and de�ne I in the same way

as in the non-limit step.

Since � = cf(�) is an uncountable cardinal, there exists n < ! such that there

is a co�nal subset C of � satisfying dI� (M) = n for each � 2 C. Hence w.l.o.g we

can assume that dI� (M) = n for each � < �:

Now we are ready to �nd a suitable in�nitely generated factor of M with the

essential socle.

Denote by �� the natural projection of M onto M=(MI�), and let � : M !Q�<�M=(MI�) be the product of the homomorphisms �� , � < �. Note thatT�<� �(M)I� =

T�<� �(MI�) =

T�<� �(ker��) = 0. As the module M is dually

slender and the module M=MI� is semisimple, M=MI� �= (R=I�)(n). Since J��

I for each � �, MJ�� MI . >From this it follows that � (MJ�� ) = 0

and �(MJ�� ) 6= �(M) for each � �. Moreover, �(M) =S�<� �(MJ�� ) =S

�<� �(M)J�� , because M =S�<�MJ�. So �(M) is the union of a strictly

increasing chain of submodules (a suitable subchain of �(M)J�� ). Consequently,

�(M) is an in�nitely generated dually slender module.

Fix an arbitrary non-zero element m 2 �(M): Let � be the minimal ordinal such

that mR 6� �(M)I� (it exists becauseT�<� �(M)I� = 0). By the construction of

sequences (�� ; � < �) and (I� ; � < �) there exists a central idempotent e 2 J��+1 n

I� : As I� is a maximal ideal, eR+I� = R, someR+mI� = mR. Hence dI� (meR) =

dI� (mR) 6= 0. Since � (meR) � � (mR) = 0 for each < � (minimality of �) and

� (meR) = 0 for each > � (indeed, eR � J��+1 � I ), 0 6= meR � ��(M): As

��(M), � < �, is semisimple, Soc(�(M)) is essential in �(M):

In addition, Soc(�(M)) is a submodule ofL

�<�M=MI� �=L

�<�(R=I�)(n). So

Soc(�(M)) is embeddable in (R�)(n): Since (R�)(n) �= (R�)n is an injective module,

�(M) is embeddable in (R�)n as well. Hence, from Lemma 2.4 it follows that R�

contains an in�nitely generated dually slender submodule. �

Theorem 2.7 shows that the implication of Proposition 1.7 cannot be reversed.

Indeed, there exist examples of non-steady commutative regular rings [EGT]. From

Proposition 1.6 it follows that any representative class of dually slender modules is

a set, but the module R� contains in�nitely generated dually slender submodules.

References

[AF] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules. 2nd edition, Springer,New York, 1992.

[B] H.Bass, Algebraic K-theory, Benjamin, New York, 1968 .[CM] R. Colpi and C. Menini, On the structure of �-modules, J. Algebra 158, 1993, 400{419.[CT] R. Colpi and J. Trlifaj, Classes of generalized �-modules, Comm. Algebra 22, 1994, 3985{

3995.


[EGT] P.C. Eklof, K.R. Goodearl and J. Trlifaj, Dually slender modules and steady rings, ForumMath., 1997, 9, 61{74.

[EM] P.C. Eklof and A.H. Mekler, Almost Free Modules, North-Holland, New York, 1990 .[G] K. R. Goodearl, Von Neumann Regular Rings, London, 1979, Pitman, Second Ed., Melbourne,

FL, 1991, Krieger.[R] R. Rentschler, Sur les modules M tels que Hom(M;�) commute avec les sommes directes,

C.R. Acad. Sci. Paris, 268, 1969, 930{933.[RTZ] P.R�u�zi�cka, J.Trlifaj and J.�Zemli�cka, Criteria of steadiness, Proc. Conf. \Abelian Groups,

Module Theory and Topology" (Padova 1997), Marcel Dekker, New York, 1998, 359{371.[S] Bo Stenstr�om, Rings of Quotients, Berlin, 1975, Springer-Verlag.[T1] J. Trlifaj, Strong incompactness for some non-perfect rings, Proc. Amer. Math. Soc. 123,

1995, 21{25.[T2] J. Trlifaj, Steady rings may contain large sets of orthogonal idempotents, Proc. Conf.

\Abelian Groups and Modules" (Padova 1994), Kluwer, Dordrecht, 1995, 467{473.[ZT] J. �Zemli�cka and J. Trlifaj, Steady ideals and rings, Rend. Sem. Mat. Univ. Padova, 1997,

98, 161{172.

Katedra algebry, MFF UK, Praha 8, Sokolovsk�a 83, 186 75, Czech Republic


D. STEADINESS OF REGULAR SEMIARTINIAN RINGS WITH

PRIMITIVE FACTORS ARTINIAN

JAN �ZEMLI�CKA

Abstract. We provide a ring-theoretic criterion of steadiness which appliesto all regular semiartinian rings with primitive factors artinian.

In the 60's, Hyman Bass remarked that the covariant functor Hom(M;�) com-

mutes with direct sums if and only if the module M is not a union of a countably

in�nite increasing chain of proper submodules [3]. Such a module has been known

and studied under various terms (�-compact, of type �, U-compact). We called it

dually slender according to [5]. As the functor Hom(M;�) is well-known to com-

mute with direct sums for each �nitely generated module M , every �nitely gener-

ated module is dually slender. Although the class of all �nitely generated modules

coincides with the class of all dually slender ones for some important classes of rings

(such as right neotherian or perfect rings), several constructions of dually slender

modules that are not �nitely generated were described. In fact, only three ways

of producing these modules are known: constructions of rings containing in�nitely

generated dually slender (right) ideals ([5], [10], [8]), constructions of a directed

system of (right) ideals whose direct limit is an in�nitely generated dually slen-

der module ([11], [13]) and constructions of rings over which all injective modules

(obviously including in�nitely generated ones) are dually slender ([4], [9]).

A general ring-theoretical characterization of rings over which there does not

exist any in�nitely generated dually slender right module, i.e. of right steady rings

is still not done. As we have noted, the question is solved only for some classes of

rings. In the present paper, we focus on the class of all regular semiartinian rings

with primitive factors artinian (for basic properties concerning this class of rings

see [2], [7], [8], [12]). The examples appeared in paper [5] show that the classes of

both steady and non-steady semiartinian regular rings are non-empty. And it is

proved in [8, Theorem 3.4.] that a semiartinian abelian regular ring R is steady if

and only if no factor of R contains an in�nitely generated dually slender ideal. The

present work generalizes this characterization for regular semiartinian rings with

primitive factors artinian.

Using the notion of a homogeneous ideal and Kaplansky's idea concerning regular

rings with primitive factors artinian [6, Theorem 7.14] we show that the structure

of a non-steady regular semiartinian ring with primitive factors artinian is not far

Research supported by grants GAUK 448/2004/B-MAT, GA�CR 201/00/P071 and by the re-search project MSM 0021620839.

53


from the structure of an abelian regular semiartinian ring (Proposition 2.7 and

Theorem 2.8). Namely, we �nd "enough" central idempotents in a suitable factor

of every non-steady ring which generate an in�nitely generated dually slender ideal

(Lemma 3.3). This allows to generalize the proof of the characterization of steadi-

ness which is done in [8, Lemmas 3.1-3.]. Finally, Theorem 3.5 says that a regular

semiartinian ring with primitive factors artinian is right steady if and only if no

factor of the ring contains any in�nitely generated dually slender right (left) ideal.

In the sequel, a ring means an associative ring with unit and a module is a

right module. A ring R is (von Neumann) regular provided that each x 2 R has a

pseudo-inverse element (i.e. there is a y 2 R satisfying xyx = x). A regular ring R

is abelian regular if all idempotents of R are central. De�ne by induction the Loewy

chain of a module M : M0 = 0, M�+1=M� = Soc(M=M�) and M� =S�<�M� for

a limit ordinal �. The Loewy length is the least ordinal � such that M� = M�+1.

We say thatM is semiartinian ifM =M�, i.e if each non-zero factor-module ofM

contains a simple submodule. Recall that a ring R is right semiartinian if RR is a

semiartinian module. A primitive factor of a ring is a factor modulo the annihilator

of a simple module. A submodule N �M is essential in M if N \K 6= 0 for every

nonzero submodule K � M . Denote by gen(M) the least cardinality of a set of

generators ofM . Note that we identify cardinals with the least ordinals of the given

cardinality.

Finally, the term ideal means a two-sided ideal. We say that an ideal is �nitely

(or in�nitely) generated if it is �nitely (or in�nitely) generated as a two-sided ideal.

1. Dually slender modules

We start this section by recalling some general facts concerning dually slender

modules and steady rings.

Theorem 1.1. (1) The class of all dually slender modules contains all �nitely

generated ones and it is closed under taking homomorphic images, extensions and

�nite sums.

(2) The class of all right steady rings is closed under taking homomorphic images

and �nite direct products.

(3) Every countably generated dually slender module is �nitely generated.

Proof. (1) See [4, Corollary 1.2].

(2) See [4, Lemma 1.9 and Lemma 2.1].

(3) See [9, Lemma 1.2] � �

It is well known (see for example [6, Chapter 6]) that both the class of all regular

semiartinian rings and the class of all rings with primitive factors artinian are closed

D. STEADINESS OF REGULAR SEMIARTINIAN RINGS 55

under taking homomorphic images and �nite direct products. Hence the assertion

of Theorem 1.1 (2) is also true for the class of all right steady regular rings with

primitive factors artinian.

Notation. According to [8] we denote by R the class of all regular semiartinian

rings with primitive factors artinian. Recall that every right module over a right

semiartinian ring is semiartinian. Take a ring R 2 R and let L = (S� j � �

� + 1) be the right Loewy chain of R. Recall that for each � � � there are a

cardinal ��, positive integers n�� , � < ��, and skew-�elds K�� , � < ��, such

that S�+1=S� �=L

�<��Mn�� (K��) (as rings without unit). The pre-image of

Mn�� (K��) coincides with the �-th homogeneous component of R=S� and it is

�nitely generated as right R=S�-module for all � < �� (cf. [8, Theorem 2.1]). The

system D(R) = f(��; f(n�� ;K��) j � < ��g) j � � �g is called the dimension

sequence of the ring R.

Finally, note that each primitive factor of R is isomorphic to a matrix ring over a

skew-�eld and so the set of all primitive ideals coincides with the set of all maximal

ideals (cf. [8, Proposition 2.6]).

Lemma 1.2. Let R 2 R and let M be an R-module. Then M = 0 if and only if

MI =M for every maximal ideal I (cf. [6, Lemma 6.14]).

Proof. If M 6= 0, there exists a simple module S � M . Then I = Ann(S) is a

maximal ideal. Since R is regular, MI \ S = 0. Thus MI 6=M . � �

Lemma 1.3. Let R 2 R and let M be a dually slender R-module. Then M=MI is

a �nitely generated semisimple module for every maximal ideal I.

Proof. M=MI is a dually slender module over a semisimple ring R=I. HenceM=MI

is semisimple and dually slender and so it is a �nitely generated module. � �

Let R be a ring. Following [6, p. 71], we de�ne the index of an ideal J (denote

it by i(J)) as the supremum of the indexes of all nilpotent elements x 2 J , i.e. of

all minimal natural numbers n such that xn = 0.

Obviously, i(S) � i(R) whenever S is either a subring or an ideal of R. Let us

remark that the index of a matrix ring Mn(T ), where T is an abelian regular ring,

is equal to n. Moreover, i(Q�<�R�) = supfi(R�)j � < �g for any arbitrary system

of rings R�, � < �.

Notation. Let M be a module and J � M(R) where M(R) denotes the set

of all maximal ideals of R. Denote by �JM : M !QI2J M=MI the product of

the natural projections �I :M !M=MI. Note that the ring-homomorphism �JR

maps R=TI2J I �= �JR(R) into the ring

QI2J R=I. Moreover, �JM (M) is a right

�JR(R)-module. Put In = fI 2M(R)j i(R=I) = ng, pnM = �InM and �n = �InR.

Since R is a semiartinian ring with primitive factors artinian, M(R) =Sn<! In.

Finally, note that �nM (M) has a naturally de�ned structure of �n(R)-module.

Lemma 1.4. Let R 2 R.


(1) Let D = f(��; f(n�� ;K��) j � < ��g) j � � �g be a dimension sequence of

the ring R. Then i(R) = supfn0� j � < �0g.

(2) If �n(R) 6= 0, then i(�n(R)) = n.

Proof. (1) By [6, Proposition 7.7. and Corollary 7.8] the index of Soc(R) is equal to

supfn0� j � < �0g. As Soc(R) is essential in R, i(R) = i(Soc(R)) = supfn0� j � <

�0g by [6, Corollary 7.5].

(2) Obviously, i(�n(R)) � n. Since there exists an ideal I 2 In such that the

homomorphism �I : R! R=I is onto, the reverse inequality holds true. � �

Proposition 1.5. If R 2 R is not a right steady ring, there exists an ideal K and

an integer n such that i(R=K) = n and R=K is not right steady.

Proof. Let M be a dually slender module. Assume that pnM (M) is a �nitely gen-

erated �n(R)-module for every n. Then there exist �nitely generated submodules

Fn � M , n < !, such that pnM (Fn) = pnM (M). Hence �I(M=P

n<! Fn) = 0

for every I 2 M(R) and M =P

n<! Fn by Lemma 1.2. We have proved that

gen(M) � !. As M is a dually slender module, it is �nitely generated by Theo-

rem 1.2 (3). Thus there exists an n such that pnM (M) is an in�nitely generated

dually slender �n(R)-module wheneverM is in�nitely generated. Finally, note that

i(�n(R)) = n by Lemma 1.4 (2) and put K = Ker �n. � �

2. 2-Dually slender ideals

An ideal J is said to be 2-dually slender provided J is not a union of a countably

in�nite strictly increasing chain of ideals, i.e. J =Sn<! Jn implies there exists an

n such that Jn = J for every increasing chain of ideals Jn, n < !, contained in J .

As we proceed to show in the following lemma, the interval h0; Ji of the lattice

of all two-sided ideals for every 2-dually slender ideal J has the same properties as

the lattice of all submodules of a dually slender module.

Lemma 2.1. Let J be a 2-dually slender ideal of a ring R.

(1) Let S be a ring and � : R ! S be a surjective homomorphism of rings.

Then �(J) is a 2-dually slender ideal of S.

(2) Let J be countably generated as a two-sided ideal. Then J is �nitely gener-

ated as a two-sided ideal.

Proof. (1) Let �(J) =Sn<!Kn for an increasing chain of ideals Kn, n < !.

Then J = J \ (Sn<! �

�1(Kn)) =Sn<!(J \ ��1(Kn)). As J is 2-dually slender,

J = J \ ��1(Kn) for some n, hence �(J) = Kn.

(2) Let fjij i < !g be a set of generators of J . Since J =Sn<!

Pi�nRjiR, we

get that J =P

i�nRjiR for a suitable n, i.e. J is a �nitely generated ideal. � �


Lemma 2.2. Let J be an ideal generated by central elements. Then the following

conditions are equivalent:

(1) J is 2-dually slender.

(2) J is dually slender as a right ideal.

(3) J is dually slender as a left ideal.

Proof. An easy exercise. � �

In the sequel, we investigate basic properties of 2-dually slender ideals of rings

belonging to the class R. It is well known that every �nitely generated ideal of

a regular ring is generated by one (idempotent) element. Therefore we focus our

attention on in�nitely generated 2-dually slender ideals.

Lemma 2.3. Let R 2 R and let J be an in�nitely generated 2-dually slender ideal.

Then there exists a natural number n such that �n(J) is an in�nitely generated

2-dually slender ideal.

Proof. The proof works similarly as the proof of Proposition 1.5. If �n(J) was

�nitely generated for every n < !, J would be countably generated, hence by

Lemma 2.1 (2) J would be �nitely generated which contradicts the hypothesis. �

�

An ideal J is said to be homogeneous of index n provided that for every I 2M(R)

the ideal �I(J) is either isomorphic to a matrix ringMn(K) for a suitable skew-�eld

K or it is equal to zero. We say that J is homogeneous if there exists a natural

number n such that J is homogeneous of index n.

Lemma 2.4. Let A and B be ideals of a regular ring R such that B � A and let n

be a natural number. Then A is a homogeneous ideal of index n if and only if both

B and A=B are homogeneous ideals of index n.

Proof. Note that (A=B)=(A=B)I = (A=B)=(AI+B=B) �= A=(AI+B) and B=BI =

B=(B \ AI) �= (B + AI)=AI for every I 2 M(R). From this immediately follows

the direct implication. Since (B + AI)=AI is an ideal of R=AI contained in A=AI

and AI is a maximal ideal in A, the reverse implication holds true. � �

The following lemma generalizes the idea of the proof of [6, Theorem 7.14].

Lemma 2.5. Let R 2 R, J a homogeneous ideal and x 2 J . Then RxR is generated

by a central idempotent.

Proof. Fix an element x 2 J and put n = i(J). Note that RxR is homogeneous by

Lemma 2.4.

Let us �rst suppose that there exist elements e; y; ri; si; gij 2 R for i; j � n such

that the following system of equations holds true:

e =X

i�n

rixsi; ey = x; ee = e;X

i�n

gii = e; gijgjl = gil; gjjgkk = 0


whenever j 6= k. Obviously, e is an idempotent and ReR = RxR. We need to

prove that e is a central element.

Fix an I 2 M(R) and suppose �I(e) 6= 0. Then there exists an i such that

�I(gii) 6= 0. Moreover �I(gjj) 6= 0 as well, since gii = gijgjjgji, for every j.

As f�I(gii)j i � ng forms an orthogonal set of non-zero idempotents, �I(e) is

an idempotent matrix of rank n. Hence it is the identity matrix, i.e. a central

idempotent. The element 0 is a central idempotent as well, so all �I(e), I 2M(R),

are central idempotents generating �I(RxR).

Let us consider the ring homomor�sm � : R !QI2M(R)R=I de�ned by the

formula �(r) = (�I(r)jI 2 M(R)). As Ker � = 0 by Lemma 1.2, e is central if

�(e) is central, which is true since �I(e) is central for every I 2 M(R). We have

proved that e is a central idempotent generating the ideal RxR.

It remains to solve the system of equations. According to [6, Lemma 6.9] it

su�ces to solve the equations in all primitive factors. The element �I(x) is either

equal to zero or �I(x) is a non-zero element of a ring of matrices of the size precisely

equal to n over a skew-�eld, I 2M(R). Since the system of the equations is easily

solvable in both the cases we get the required central idempotent. � �

Lemma 2.6. Let n be a natural number and let J be an ideal of a ring R 2 R.

Suppose that �j(J) = 0 whenever j 6= n. Then J is homogeneous of index n.

Proof. Take an arbitrary maximal ideal I 2 M(R). Suppose that �I(J) is non-

zero, i.e. �I(J) = R=I and put k = i(R=I). Note that as the module �I(J) is a

non-zero subfactor of �k(J), �k(J) 6= 0. By the premise we get that k = n, so J is

homogeneous of index n. � �

Proposition 2.7. Let R 2 R and let J be an in�nitely generated 2-dually slender

ideal. Then there exists a factor ring S of R (denote the natural projection R! S

by p) such that p(J) is an in�nitely generated 2-dually slender homogeneous ideal.

Proof. By Lemma 2.3 there exists a natural number n such that �n(J) is an in-

�nitely generated ideal. Take n minimal. Then i(�n(R)) = n by Lemma 1.4 (2).

Now, �j(J) is a �nitely (in fact one-) generated ideal for every j < n, hence there

exists an element y 2 J such that �j(J) = �j(RyR) for each j < n. Thus the ring

S = �n(R=RyR) contains the in�nitely generated ideal K = �n(J=RyR) which is

2-dually slender by Lemma 2.1 (1). Moreover, pjS(S) = 0 and so pjS(K) = 0 if

j > n because i(S) � n. Finally, pjS(K) = 0 if j < n, since �j(J) = �j(RyR).

Applying Lemma 2.6 we obtain the required result. � �

Theorem 2.8. Let R be a regular semiartinian ring with primitive factors artinian

such that at least one factor of R contains an in�nitely generated 2-dually slender

ideal. Then R is neither right nor left steady. Moreover, there exists a factor of R

containing an in�nitely generated dually slender right (left) ideal.

Proof. By Proposition 2.7 there exists a factor ring S of R containing an in�nitely

generated 2-dually slender ideal J which is generated by central idempotents. From


Lemma 2.2 it follows that J is in�nitely generated and dually slender as both a

right and a left ideal. � �

3. Steadiness

Let us �rst recall and extend [8, De�nition 2.7]. Let L = (S� j � � �+1) be the

right Loewy chain of R and let M 2 ModR. Let P = fP�� j � � �; � < ��g be a

representative set of the isomorphism classes of all simple modules.

(1) For each � � �, MS�+1=MS� is called the �-th slice of M .

(2) An ordinal � is said to be the height of M (denote it by hR(M)) if it is the

minimal � � � + 1 such that MS� =M .

(3) M is said to be saturated provided that P�� is a subfactor of M for all

� < hR(M) and all � < ��. Let J be an ideal. Then M is said to be

J-saturated if P�� is a subfactor of M for those � < hR(M) and � < ��

for which P�� is a subfactor of J .

Suppose (J� j � � �) is the right Loewy chain of an ideal J . Note that by

regularity J� = J \ S� = JS�, i.e. the �-th slice of J is isomorphic to J�+1=J�.

Moreover, the �-slice of any module MJ is isomorphic to MJ�+1=MJ�.

Lemma 3.1. Let R 2 R be of uncountable socle length. Let I be the class of all

in�nitely generated dually slender modules. Assume I 6= ; and take M 2 I such

that the ordinal h �R(M) is minimal, where �R = R=AnnR(M). Then h �R(M) is a

limit ordinal of co�nality � !1.

Proof. See [8, Lemma 3.1] � �

Lemma 3.2. Let R 2 R, M , a dually slender module and J , an ideal such that

M =MJ . Suppose that M is J-saturated. Then J is 2-dually slender.

Proof. Let J =Sn<! In for an increasing chain of ideals In. We will follow the

proof of [8, Lemma 3.3] where we replace S� by J .

As M = MJ =Sn<!MIn is dually slender, there exists an m < ! such that

MIm = M . By [8, Proposition 2.6], if Im 6= J then there are � < hR(M) and

� < �� such that Im � Ann(P��) and P�� is a subfactor of J . Moreover, P�� is

not a subfactor of M by [8, Lemma 2.8(i)] which contradicts the assumption of M

being J-saturated. Hence Im = J . � �

Lemma 3.3. Suppose that R 2 R is not right steady. Then there exist a factor S of

the ring R, an in�nitely generated dually slender S-module M and a homogeneous

ideal J � S such that MJ =M .

Proof. Fix an in�nitely generated dually slender module M such that the ordinal

� = h �R(M) is minimal, where �R = R=AnnR(M). By Proposition 1.5 we may

suppose that R is of a bounded index and �R = R. Denote by L the �-th member of


the Loewy chain of R. Then ML =M . Take a minimal n such that pnM (M) is an

in�nitely generated module. We may suppose that pkM (M) = 0 for every k < n.

Indeed, since pkM (M) is �nitely generated for every k < n, there exists a �nitely

generated module F such that pkM (M) = pkM (F ), hence pkM=F (M=F ) = 0 for

every k < n. As pnM (M) is a �n(R)-module we may replace the module M by

pnM (M) and the ring R by �n(R). Clearly, i(R) = n. Put J = L\ (Tk<nKer �k)

Note that �k(R) = 0 for every k � n and �k(J) = 0 for every k < n. Thus the

ideal J is homogeneous of index n by Lemma 2.6. Moreover, Soc(L=J) contains

no homogeneous component of index n by Lemma 2.4. Hence i(Soc(L=J)) < n.

Applying [6, Corollary 7.5] we get i(L=J) = i(Soc(L=J)) < n. Put �M = M=MJ

and �L = L=J . As i(�L) < n, pnR=J(�L) = 0. Thus pn �M ( �M) = pn �M ( �M �L) =

pn �M ( �M)�nR=J(�L) = 0. Moreover, pk �M ( �M �L) = 0 for each k < n, hence �M = 0 and

M =MJ . Recall that J is homogeneous which �nishes the proof. � �

Lemma 3.4. Assume R 2 R is not right steady. Then there exist a factor ring

S of R, an in�nitely generated ideal J � S and a J-saturated in�nitely generated

dually slender S-module M such that MJ =M .

Proof. By Lemma 3.3 there exists an in�nitely generated dually slender S-module

M and a homogeneous ideal J � S such that MJ = M where S is a suitable

factor-ring of R. Moreover, we may suppose that Ann(M) = 0 and that the

height of M (and so, the Loewy length of J) is minimal. Denote it by �. Let

(J�j � � �) be the right Loewy chain of J . Since J is homogeneous, every central

idempotent generating a homogeneous component modulo J� can be lifted to a

central idempotent by Lemma 2.5. Now, following the proof of [8, Lemma 3.2] we

show by induction on � < � that a simple module P embeds into the �-th slice of

M if P embeds into J=J�.

Put A = Ann(Soc(M)). AsM 2ModR is faithful,MA\Soc(M) = 0, so A = 0.

Hence Soc(M) contains a copy of each simple module P � J0 = Soc(J) and the

assertion is true for � = 0.

Take 0 < � < � and assume that a simple submodule P of the �-th slice

of J does not embed into the �-th slice of M . We have proved that there is a

central idempotent e 2 R such that e + J� generates homogeneous component

corresponding to P .

If � is non-limit, we apply [8, Proposition 2.6] to get that M 0 = (M=MJ��1)e is

a dually slender submodule of the (��1)-th slice ofM . AsM 0 is �nitely generated,

there is an idempotent f 2 J� such that M 0f =M 0. Thus Me(1� f)R �MJ��1.

By the induction premise for ��1 and by [8, Proposition 2.6], we get e(1�f) 2 J��1,

whence e 2 J�, a contradiction.

If � is a limit ordinal, we have Me � MJ�, whence Me = MeJ�. Since Me is

dually slender, the minimality assumption implies that Me is �nitely generated, so

Me � MJ for some < �. By the induction premise for and by [8, Proposi-

tion 2.6], we get e 2 J , a contradiction. � �


Theorem 3.5. Let R 2 R. Then the following conditions are equivalent:

(i) R is not right steady;

(ii) R is not left steady;

(iii) There exists an in�nitely generated (as a two-sided ideal) 2-dually slender

ideal of a suitable factor-ring of R.

(iv) There exists an in�nitely generated dually slender right ideal of a suitable

factor-ring of R.

(v) There exists an in�nitely generated dually slender left ideal of a suitable

factor-ring of R.

Proof. The condition (iii) is left-right symmetric, therefore it su�ces to prove the

equivalence of conditions (i), (iii) and (iv).

(i)! (iii) It follows from Lemmas 3.1, 3.2 and 3.4.

(iii)! (iv) It is proved in Theorem 2.8.

(iv)! (i) Trivial. � �

References

[1] F.W. Anderson and K.R. Fuller Rings and Categories of Modules, 2-nd edition, New York1992, Springer.[2] G. Baccella Semiartinian V-rings and semiartinian von Neumann regular rings, J. Algebra173 (1995), 587{612.[3] H. Bass Algebraic K-theory, New York 1968, Benjamin.[4] R. Colpi and J. Trlifaj Classes of generalized �-modules, Comm. Algebra 22 (1994), 3985{3995.[5] P.C. Eklof, K.R. Goodearl and J. Trlifaj Dually slender modules and steady rings, ForumMath. 9 (1997), 61{74.[6] K. R. Goodearl Von Neumann Regular Rings, London 1979, Pitman, Second Ed. Melbourne,FL 1991, Krieger.[7] C. N�ast�asescu and N. Popescu, Anneaux semi-artiniens, Bull. Soc. Math. France 96 (1968),357{368.[8] P.R�u�zi�cka , J. Trlifaj and J. �Zemli�cka Criteria of steadiness, Abelian Groups, Module Theory,and Topology, New York 1998, Marcel Dekker, 359{372.[9] J. Trlifaj Almost �-modules need not be �nitely generated, Comm. Algebra, 21 (1993), 2453{2462.[10] J. Trlifaj Steady rings may contain large sets of orthogonal idempotents, Abelian groups andmodules, Proc. conf. Padova, Italy, June 23-July 1, 1994, Dordrecht 1995, Kluwer 467{473.[11] J. �Zemli�cka !1-generated uniserial modules over chain rings, Comment. Math. Univ. Caroli-

nae, 45 (2004), 403{415.[12] J. �Zemli�ckaWhich regular semiartinian rings with primitive factors artinian exist?, preprint,2004.[13] J. �Zemli�cka and J. Trlifaj, Steady ideals and rings, Rend. Sem. Mat. Univ. Padova, 98 (1997),161{172.

Katedra algebry MFF UK, Sokolovsk�a 83, 186 75 Praha 8, �Cesk�a Republika, zem-

[email protected]

Chapter 4

The defect functor of

homomorphisms and direct

unions

This chapter is constituted by the single article devoted to commuting propertiesof the defect functor which generalizes the notions of Hom and Ext functors:

E. Simion Breaz, Jan �Zemli�cka, The defect functor of homomorphisms and

direct unions, Algebr. Represent. Theor. 19/1 (2016), 181{208.

62

E. THE DEFECT FUNCTOR OF A HOMOMORPHISM AND

DIRECT UNIONS

SIMION BREAZ AND JAN �ZEMLI�CKA

Abstract. We will study commuting properties of the defect functor Def� =CokerHomC(�;�) associate to a homomorphism � in a �nitely presented cat-egory. As an application, we characterize objects M such that Ext1

C(M;�)

commutes with direct unions (i.e. direct limits of monomorphisms), assumingthat C has a generator which is a direct sum of �nitely presented projectiveobjects.

1. Introduction

Commuting properties of some canonical functors de�ned on some categories

play important roles in the study of various mathematical objects. For instance,

�nitely presented objects in a category with directed colimits are de�ned by the

condition that the induced covariant Hom-functor commutes with all directed col-

imits. In the case of module categories the equivalence between the property used

in this de�nition and the classical notion of �nitely presented module was proved

by Lenzing in [26]. In that paper it is also proved that there are strong connections

between commuting properties of covariant Hom-functors and commuting proper-

ties of tensor product functors with respect to direct products. These connections

were extended to the associated derived functors in [9] and [12]. Moreover, Drinfeld

proposed in [15] to use at Mittag-Le�er modules in order to construct a theory

for in�nite dimensional vector bundles. Recent progresses in this directions were

obtained in [8], [17] and [18]. Auslander introduced in [6] the class of coherent

functors, and W. Crawley-Boevey characterized (in the case of module categories)

these functors as those covariant functors which commute with direct limits and

direct products, [14, Lemma 1]. This result was extended to locally �nitely pre-

sented categories by H. Krause, [24, Chapter 9]. The in uence of these functors is

presented in [14] and [20].

Brown [12] and Strebel [32] used commuting properties of covariant Ext1C-functors

with respect to direct limits in order to characterize groups of type (FP). In module

theory an important ingredient used in the study of tilting classes (e.g. [19, Lemma

5.2.18 and Theorem 5.2.20]) is a homological characterization, [19, Theorem 4.5.6],

2000 Mathematics Subject Classi�cation. 18G15, 16E30, 16E05.Key words and phrases. defect functor, coherent functor, direct limit, direct union, Ext-

functor.Research supported by the CNCS-UEFISCDI grant PN-II-RU-PCE-2012-4-0100.

63


of the closure lim�!C, where C is a class of FP2-modules. This is based on the fact

that Ext1R(M;�) commutes with direct limits whenever M is an FP2-module, [19,

Lemma 3.1.6]. In the case of Abelian groups, commuting properties of Ext1 functors

with respect to particular direct limits were also studied in [4] and [31].

In this paper we will focus on commuting properties with respect to direct limits

for the defect functor associated to a homomorphism in a locally �nitely presented

abelian category. Let us introduce basic notions which we will use in the sequel. Let

M be an object in an additive category C with directed colimits, and G : C ! Ab

a covariant functor. Furthermore suppose that F = (Mi; vij)i;j2I is a directed

system of objects in C such that there exists lim�!Mi and let vi :Mi ! lim�!Mi be the

canonical homomorphisms. Then (G(Mi); G(vij)) is also a direct system, and we

denote by lim�!G(Mi) its direct limit. Moreover, we have a canonical homomorphism

�F : lim�!G(Mi)! G(lim�!Mi)

induced by the homomorphisms G(vi) : G(Mi)! G(lim�!Mi), i 2 I.

We say that G commutes with F if �F is an isomorphism. The functor G com-

mutes with direct limits (direct unions, resp. direct sums) if the homomorphisms �F

are isomorphisms for all directed systems F (such that all vij are monomorphisms,

resp. all direct sums).

Let C be an additive category with direct limits. We recall from [1] and [2] that

an object M is �nitely presented (�nitely generated) respectively if and only if

HomC(M;�) commutes with direct limits (of monomorphisms), i.e. the canonical

homomorphisms

MF : lim�!HomC(M;Mi)! HomC(M; lim�!Mi)

are isomorphisms for all direct systems F = (Mi; vij) (such that all vij are monomor-

phisms). The category C is �nitely accessible if C has directed colimits and every

object is a direct limit of �nitely presented objects. A cocomplete �nitely accessible

category C is a locally �nitely presented category.

The notion of defect functor associated to a homomorphism extends the defect

functor of an exact sequence used in [7]. This functor represents generalizations for

the following canonical functors: the Hom-covariant functor induced by an object,

the Pext-covariant functor induced by an object, respectively the Ext1-covariant

functor in the case when C is a functor category.

In Section 2 we introduce the defect functor Def� : C ! Ab associated to a

homomorphism �, and we establish some basic properties for this functor. In

Theorem 3 we show that the canonical decomposition of � induces a short exact

sequence of defect functors. Since Def� commutes with direct products, we can

apply [14] and [25] to manage the case when Def� commutes with all direct limits.

Therefore we will focus our study to commuting properties with respect to particular

direct limits.

E. DEFECT FUNCTOR AND DIRECT UNIONS 65

In Section 3 we study when the natural homomorphism ��F: lim�!Def�(Mi) !

Def�(lim�!Mi); where F = (Mi; vij)i;j2I is a directed family in C, is an epimorphism.

It is proved that ��Fis an epimorphism for all directed family F (of monomorphisms)

if and only if � is a section in the quotient category of C modulo the ideal of all

homomorphisms which factorizes through a �nitely presented (generated) object.

We apply these results in Sections 4 and 5 in order to characterize the homomor-

phisms � such that Def� commutes with direct unions or direct sums. Assuming

that there is no !-measurable cardinal we prove that it is enough to consider only

commuting of Def� with countable direct sums (Proposition 26).

In Section 6 (this section includes the results proved in the unpublished man-

uscript [11]) we apply the previous results to characterize objects M in a functor

category with the property that the functor Ext1C(M;�) commutes with direct

unions (Theorem 44). These are exactly the direct summands in direct sums of

projective objects and �nitely presented objects. In [15, Section 6] the author used

these objects (called, 2-almost projective modules) in order to study various kind

of objects, e.g. di�erentially nice k-schemes are de�ned using 2-almost projective

modules. These objects are also studied in [21] for the case of quasivarieties, cf.

[21, Proposition 4.3].

For the case of coherent categories these are exactly those objects such that the

induced Ext1C-covariant functor commutes with direct limits (Corollary 47). We

mention that in fact the structure of these objects can be very complicated. For

such an example we refer to [27, Lemma 4.3].

Furthermore, Theorem 34 gives a description of objectsM for which Ext1C(M;�)

commutes with direct sums using some splitting properties of projective presenta-

tions of M . We close the paper with a discussion about steadiness relative to

Ext1, i.e. the condition when commuting of Ext1(M;�) with direct sums implies

commuting of Ext1(M;�) with direct unions.

In this paper C will denote an locally �nitely presented abelian category, i.e.

C is a Grothendieck category with a generating set of �nitely presented objects.

Therefore, an object is �nitely generated i� it is an epimorphic image of a �nitely

presented object [1, Proposition 1.69], and the structural homomorphisms associ-

ated to direct unions are monomorphisms by [1, Proposition 1.62].

2. The defect functor associated to a homomorphism

In order to de�ne the defect functor Def� associated to a homomorphism � it

is useful to consider, as in [24], the big category (C; Ab) of all additive covariant

functors from C into the category of all abelian groups. Albeit (C; Ab) is not a

category we can construct pointwise all notions which de�ne abelian categories

(kernels, cokernels, direct sums etc.), and the universal properties associated to

these notions can be transfered from Ab to (C; Ab). For instance, if � : F ! G

is a natural transformation then we can de�ne a functor Coker(�) and a natural


transformation � : G ! Coker(�) in the following way: For all X 2 C we de�ne

Coker(�)(X) = Coker(�X) = G(X)=Im(�X), and for every � : X ! Y we de�ne

Coker(�)(�) : Coker(�)(X) ! Coker(�)(Y ) is the unique map which make the

diagram

F (X)

F (�)

��

�X// G(X)

�X//

G(�)

��

G(X)=Im(�X)

Coker(�)(�)

��

// 0

F (Y )�Y// G(Y )

�Y// G(Y )=Im(�Y ) // 0

commutative, where �X : G(X) ! Coker(�)(X) and �Y : G(Y ) ! Coker(�)(Y )

are the canonical epimorphisms. It is not hard to see that Coker(�) is a functor,

and the collection �X de�ne a natural transformation G! Coker(�) which has the

same universal property as those which de�nes the classical cokernel in additive

categories.

De�nition 1. Suppose that � : L! P is a homomorphism in C. Then � induces

a natural transformation Hom(�;�) : Hom(P;�)! Hom(L;�). The functor

Def�(�) = Coker(Hom(�;�))

will be called the defect functor associated to �.

It is clear from the previous observation that Def� is characterized by the con-

ditions:

(i) Def�(X) = Hom(L;X)=Im(Hom(�;X)) for each object X and

(ii) Def�( )(� + BX) = � + BY for each objects X;Y and homomorphisms

2 Hom(X;Y ), � 2 Hom(K;X) where BX = Im(Hom(�;X)) and BY =

Im(Hom(�; Y )).

In fact, if f : X ! Y is a homomorphism then we have a commutative diagram:

Hom(P;X) ��! Hom(L;X) ��! Def�(X) ��! 0??y

??y??y

Hom(P; Y ) ��! Hom(L; Y ) ��! Def�(Y ) ��! 0:

Here are some examples:

Example 2. Let � : L! P be a homomorphism in C.

(1) If C is abelian, P is projective and � a monomorphism, then Def�(�) is

canonically equivalent to Ext1(P=�(L);�).

(2) If P = 0, then Def�(�) is canonically equivalent to Hom(L;�).

(3) If � is an epimorphism and � : K ! L is the kernel of � then Def�(�)

represents the covariant defect functor associated to the exact sequence

0! K�! L

�! P ! 0, [23].

(4) If R is a unital ring, C = Mod-R, and L and P are �nitely generated and

projective then Def�(R) represents the transpose of P=�(L).


In the following we will prove some general properties of defect functors. Since

in the category of all abelian groups the direct products are exact, it is easy to

see that Def� commutes with direct products. Moreover, in many situations the

study of these functors can be reduced to the study of defect functors associated

to monomorphisms or to epimorphisms.

Theorem 3. Let � : L ! P be a homomorphism in the abelian category C. If

iK : K ! L is the kernel of �, �K : L ! L=K is the canonical epimorphism, and

� : L=K ! P is the homomorphism induced by � then there exists a canonical

exact sequence of functors and natural transformations

0! Def� ! Def� ! Def�K ! 0:

Proof. Starting with the exact sequence

0! K�K! L

�! P !M ! 0;

where M is the cokernel of �, we obtain the short exact sequences

0! K�K! L

�K! L=K ! 0

and

0! L=K�K! L!M ! 0:

Passing to the Hom covariant functors induced by the objects involved in the

previous exact sequences we obtain, using the Ker-Coker Lemma, the following

commutative diagram of functors and natural transformations:

0

��

0

��

0 // (M;�) // (P;�)��

// (L=K;�) //

��

Def�(�) //

��

0

0 // (M;�) // (P;�)��// (L;�) //

��

Def�(�) //

��

0

Def�K

��

Def�K

��

0 0 ;

hence the statement of the theorem is proved. �

Proposition 4. If � : L ! P is a homomorphism in C, the following statements

are true:

(1) If P is projective and M = Coker(�) then every exact sequence

0! X ! Y ! Z ! 0


induces an exact sequence

0! (M;X)! (M;Y )! (M;Z)! Def�(X)! Def�(Y )! Def�(Z):

(2) If L is projective then Def� preserves the epimorphisms.

(3) If L and P are projective then Def� is a right exact functor.

Proof. Let 0 ! X ! Y ! Z ! 0 be an exact sequence. Applying the Hom-

functors we obtain the following commutative diagram

0

��

0

��

0

��

0 // (M;X)

��

// (M;Y )

��

// (M;Z)

��

0 // (P;X)

��

// (P; Y )

��

// (P;Z)

��

0 // (L;X)

��

// (L; Y )

��

// (L;Z)

��

Def�(X)

��

// Def�(Y )

��

// Def�(Z)

��

0 0 0 ;

and the statements are obvious. �

Remark 5. Recently the defect functor associated to a homomorphism between

projective object was involved in the study of silting modules, [5]: a homomorphism

� : L ! P with L and P projective objects is a silting module if Gen(P=�(L)) =

Ker(Def�).

3. The defect functor and direct limits

Throughout the section we suppose that L�! P

�!M ! 0 is an exact sequence

in C, F = (Mi; vij)i;j2I is a direct system of objects in C, and vi :Mi ! lim�!Mi are

the canonical homomorphisms. Furthermore, we denote by

��F: lim�!Def�(Mi)! Def�(lim�!Mi)

the natural homomorphisms induced by the families Def�(vij), i; j 2 I, and Def�(vi),

i 2 I. Following the general de�nition considered in Section 1, we say that Def�(�)

commutes with F if ��Fis an isomorphism. The functor Def�(�) commutes with

direct limits (direct unions, resp. direct sums) if the homomorphisms ��Fare iso-

morphisms for all directed systems F (such that all vi are monomorphisms, resp.

all direct sums).


We have the following useful commutative diagram

(D1)

0 // lim�!(M;Mi) //

MF��

lim�!(P;Mi) //

PF��

lim�!(L;Mi)

LF��

lim�!

�i// lim�!Def�(Mi) //

��F

��

0

0 // (M; lim�!Mi) // (P; lim�!Mi)��// (L; lim�!Mi)

�// Def�(lim�!Mi) // 0

whose rows are exact, where the natural homomorphisms XF are de�ned in Section

1.

Using this diagram we have the following simple consequences:

Corollary 6. (1) If L is �nitely presented and � : L ! P is a homomor-

phism, then for every direct family F the canonical homomorphism ��Fis

an epimorphism.

(2) If L is �nitely generated and � : L! P is a homomorphism, then for every

direct family of monomorphisms F the canonical homomorphism ��Fis an

epimorphism.

Example 7. There exists a homomorphism � : L ! P and a direct family F (of

monomorphisms) such that L is �nitely presented (generated) and the canonical

homomorphism ��Fis not an isomorphism.

Proof. Let C be the category of all abelian groups. If p is a prime number we denote

by Zp = fmpkj m 2 Z; k 2 Ng � Q. If � : Z! Zp is the canonical inclusion in the

category of all abelian groups then for every torsion-free abelian group A we have

a natural isomorphism

Def�(A) �= A=Dp(A);

where Dp(A) is the maximal p-divisible subgroup of A.

We can write the abelian group Q as a union of a chain of cyclic subgroups

Fn = 1n!Z, n 2 N

�, where the connecting homomorphisms um;n : Fm ! Fn, m < n,

are the inclusion maps. Since Hom(Zp; Fn) = 0 for all n > 0, it follows that we

can identify Def�(Fn) = Fn and Def�(um;n) = um;n for all m;n 2 N�. Then

lim�!Def�(Fn) = Q. But Def�(lim�!Fn) = Def�(Q) = 0, hence ��F: Q ! 0 is not a

monomorphism. �

We will use the following lemma:

Lemma 8. An object M is �nitely generated if and only if there exists an exact

sequence 0! L! P !M ! 0 with P a �nitely presented object.

Consequently, if M is �nitely generated then for every direct system F the natural

homomorphism MF is a monomorphism. Moreover, M is �nitely presented if and

only if L is �nitely generated.


Proof. The �rst part is proved in [1, Proposition 1.69], while for the other state-

ments we can apply Ker-Coker Lemma on diagram (D1). �

Applying the above de�nitions to the diagram (D1), it is not hard approach that

case when Hom(P;�) commutes with direct sums, direct unions, respectively direct

limits. We recall that P is called small if Hom(P;�) commutes with direct sums.

Proposition 9. Let � : L! P be a homomorphism.

(1) Suppose that P is a small object. The functor Def� commutes with direct

sums if and only if L is a small object.

(2) Suppose that P is a �nitely generated object. The functor Def� commutes

with direct unions if and only if L is �nitely generated.

(3) Suppose that P is a �nitely presented object. Then Def� commutes with

direct limits if and only if L is �nitely presented.

Proof. (1) Let F = (Mi)i2I be a family of objects in C. We construct a diagram

(D1) induced by the direct sum of F. Since the class of small objects is closed with

respect to epimorphic images, MF and P

F are isomorphisms. Therefore ��Fis an

isomorphism if and only if LF is an isomorphism. The conclusion is now obvious.

(2) The proof follows the same steps as for (1), using this time a direct system

F = (Mi; �ij)i;j2I such that all �ij are monomorphisms, and the fact that the class

of �nitely generated objects is closed with respect to epimorphic images.

(3) Suppose that Def� commutes with direct limits. By what we just proved L is

�nitely generated, hence M is �nitely presented. Therefore, for every direct system

F = (Mi; �ij)i;j2I the homomorphisms MF and P

F are isomorphisms. Therefore

LF is an isomorphism, hence L is �nitely presented.

Conversely, the objects L, M , and P are �nitely presented, hence the �rst three

vertical maps in diagram (D1) are isomorphisms. Then ��Fis also an isomorphism.

�

Using the statement (2) in the above proposition we can reformulate the charac-

terization presented in [14, Lemma 1] for the case of direct unions. Since the proof

is verbatim to Crawley-Boevey's proof, it is omitted.

Theorem 10. A functor F : C ! Ab commutes with respect direct products and

direct unions if and only if it is naturally isomorphic to a defect functor Def�

associated to a homomorphism � : L! P with L and P �nitely generated.

Using the same techniques as in [10], it is not hard to see that when L and P

are projective the three commuting properties considered in the Proposition 9 are

equivalent.

Proposition 11. Let � : L ! P be a homomorphism between projective right

R-modules. Then the following are equivalent:

(1) Def� commutes with direct limits;


(2) Def� commutes with direct unions;

(3) Def� commutes with direct sums;

(4) Def� commutes with direct sums of copies of R;

Under these conditions Def�(R) is a �nitely presented left R-module.

Proof. (4))(1) From Proposition 4 and from the proof of Watts's theorem [34,

Theorem 1], we obtain that Def�(�) is naturally isomorphic to � R Def�(R).

Therefore it preserves direct limits.

Moreover, if these equivalent conditions are satis�ed the functor �R Def�(R)

preserves direct products. This is true exactly if the left R-module Def�(R) is

�nitely presented. �

It is well known that if G : C ! Ab is an additive functor then for every family

F = (Mi)i2I the natural homomorphism �i2IG(Mi)! G(�i2IMi) is a monomor-

phism. Therefore in the above proposition it is enough to verify whether the natural

homomorphisms ��Fare epimorphisms.

In the following we will study the case when the natural homomorphisms ��Fare

epimorphisms.

Lemma 12. Let � : L ! P be a homomorphism, F = (Mi; vij)i;j2I a direct

system, and let f : L ! lim�!Mi be a homomorphism. Using the same notations as

in diagram (D1), the following are equivalent:

(1) �(f) 2 Im(��F);

(2) there exists k 2 I, h : L!Mk and g : P ! lim�!Mi such that f = g�+ vkh.

Proof. The homomorphisms from (2) can be represented in the following diagram

L�//

h

{{xxxxx

f

��

P

g||zzzz

Mk vk// lim�!Mi :

(1)) (2) If we look at the commutative diagram (D1), we observe that �(f) 2

Im(��F) if and only if there is an element x 2 lim�!HomC(L;Mi) such that �(f) =

��F(lim�!�i)(x) = �L

F(x). Then f � LF(x) = ��(g) = g� for some element g 2

HomC(P; lim�!Mi).

Since x 2 lim�!HomC(L;Mi), there exist k 2 I and h 2 HomC(L;Mk) such that

x = vk(h) 2 Imvk, where vk : Hom(L;Mk) ! lim�!HomC(L;Mi) is the structural

homomorphism associated to the direct limit. Since �Fvk = HomC(L; vk), it follows

that �F(x) = HomC(L; vk)(h) = vkh. Thus f = g� + vkh.

(2))(1) If f = g� + vkh then f � vkh 2 Im(��), hence

�(f) = f + Im(��) = vkh+ Im(��) = �LF(vk(h)) = �F(lim�!�i)(vk(h));

and the proof is complete. �


In the following FP will be the ideal in C consisting of those homomorphisms

which factorize through a �nitely presented object, i.e. FP represents the collection

of subgroups FP(A;B) � HomC(A;B), A;B 2 C, of those homomorphisms A! B

which factorize through a �nitely presented object. Then C=FP will denote the

quotient category which has the same objects as C and

HomC=FP(A;B) = HomC(A;B)=FP(A;B):

In the following assertion, if f : A! B and h : A! C are homomorphisms, we

will denote by (f; h)t : A! B�C the canonical homomorphism induced by f and

h.

Theorem 13. Let � : L! P be a homomorphism. The following are equivalent:

(1) for every direct system F the map ��Fis an epimorphism;

(2) there exists g : P ! L such that 1L � g� factorizes through a �nitely

presented object;

(3) the induced homomorphism � in C=FP is a section;

(4) there exists a homomorphism h : L! F such that F is a �nitely presented

object and the induced map (�; h)t : L ! P � F is a splitting monomor-

phism.

Proof. (1))(2) We can write L as a direct limit of �nitely presented objects, L =

lim�!Li. Then an application of Lemma 12 for f = 1L gives us the conclusion.

(2))(1) Since 1L�g� factorizes through a �nitely presented object, there exists

a �nitely presented object F and two homomorphisms h1 : L ! F , h2 : F ! L

such that 1� g� = h2h1:

Since F is �nitely presented, for every direct limit lim�!Mi and every homomor-

phism f : L ! lim�!Mi we can �nd an index i and a homomorphism fi : F ! Mi

such that fh2 = vifi. It follows that f(1 � g�) = fh2h1 = vifih1. Then there

exists g0 = fg : P ! lim�!Mi and h = fih1 : L ! Mi such that f = g0� + vih, and

we apply Lemma 12 to complete the proof.

(2),(3) This is obvious.

(2))(4) Let g be as in (2) and h = 1L � g�. There exists a �nitely presented

object F and two maps h1 : L ! F , h2 : F ! L such that h = h2h1. Then the

map (�; h1)t : L ! P � F induced by � and h1 is a splitting monomorphism, and

a left inverse is the homomorphism (g; h2) : P � F ! L induced by g and h2.

(4))(2) Let g0 : P �F ! L be a left inverse for (�; h)t. Then 1L = gjP�+ gjFh,

hence 1L � gjP� factorizes through a �nitely presented object. �

4. Commuting with direct unions

Recall from [1, Proposition 1.62] that in our hypotheses the structural maps

vi :Mi ! lim�!Mi of a direct union are monomorphisms.


Since the class of �nitely generated objects is closed with respect to epimor-

phic images, we will prove that Theorem 13 can be improved to characterize the

commuting of Def� with direct unions.

Theorem 14. Let � : L! P be a homomorphism. The following are equivalent:

(1) for every direct system F of monomorphisms the induced homomorphism

��Fis an epimorphism;

(2) there exists g : P ! L such that 1L � g� factorizes through a �nitely

generated object;

(3) if FG is the ideal of all homomorphisms which factorize through a �nitely

generated object then the induced homomorphism � in C=FG is a section;

(4) there exists a homomorphism h : L ! M such that h factorizes through

a �nitely generated object and the induced map (�; h)t : L ! P �M is a

splitting monomorphism.

(5) there exists a homomorphism h : L! F such that F is a �nitely generated

object and the induced map (�; h)t : L ! P � F is a splitting monomor-

phism,

(6) there exists a �nitely generated subobject H � L such that the induced

homomorphism � : L=H ! P=�(H) is a split mono and there exists a left

inverse for � which can be lifted to a homomorphism P ! L.

Proof. It is enough to prove the equivalence (2),(6) since for the other equivalences

we can repeat the arguments of the proof of Theorem 13, using the fact that L can

be written as a direct union of its �nitely generated subobjects.

(2))(6) By (2) we know that there exists a homomorphism g : P ! L such

that 1L � g� factorizes through a �nitely generated object. Therefore there exists

a �nitely generated subobject H � L such that Im(1L � g�) � H. If h : H ! L is

the embedding of H in L then there exists a homomorphism : L! H such that

1L � g� = h .

Since h = g�h+ h h, we have Im(g�h) � Im(h), hence g�(H) � H. Therefore

there are canonical homomorphisms � : L=H ! P=�(H) and g : P=�(H) ! L=H

which are induced by �, respectively g, and the diagram

L�

��! Pg

��! L

�H

??y ��(H)

??y �H

??y

L=H�

��! P=�(H)g

��! L=H

is commutative, where the vertical arrows are the canonical epimorphisms.

Moreover, �Hh = 0, hence g��H = �Hg� = �H(1L � h ) = �H . Since �H is an

epimorphism we have g� = 1K=Kk , hence � is a splitting monomorphism.

(6))(2) Let g : P ! L be a homomorphism such that g(�(H)) � H and the

induced homomorphism g : P=�(H)! L=H satis�es the equality g� = 1L=H . Then

Im(1L � g�) � H, and the proof is complete. �


If P is projective the lifting condition stated in (6) is always satis�ed. This is

not the case if P is not projective.

Example 15. Let C be the category of all abelian groups, and let Zp be the

subgroup of Q de�ned in Example 7.

If � : Zp ! Q is the inclusion map then the induced homomorphism � : Zp=Z!

Q=Z is split mono. But Hom(Q;Zp) = 0, so the left inverse of � (in this case this

left inverse is unique) cannot be lifted to a homomorphism Q! Zp.

We obtain the following interesting characterization of pure-projective objects.

Let us recall that an exact sequence 0 ! A ! B ! C ! 0 is pure (and B ! C

is a pure epimorphism) if all �nitely presented objects are projective with respect

to it, and an object is projective if and only if it is projective with respect to

all pure exact sequences. It is not hard to see that an object is pure-projective

i� it is a direct summand of a direct sum of �nitely presented objects. As in the

standard homological algebra we can de�ne the functor Pext1(M;�) as Def� , where

� : P !M is a pure epimorphism such that P is pure-projective. Remark that M

is pure-projective i� Pext1(M;�) = 0. For more details we refer to [22, Appendix

A].

Proposition 16. The following are equivalent for an object M 2 C:

(1) The functor Pext1(M;�) commutes with direct limits;

(2) Pext1(M;�) commutes with direct unions;

(3) M is pure projective.

Proof. It is enough to prove that (2))(3).

Let M be an object such that Pext1(M;�) commutes with direct unions. Since

M is a direct limit of �nitely presented objects, there exists a pure exact sequence

0! L�! �i2IPi !M ! 0

such that all Pi are �nitely presented objects. Hence Pext1(M;�) = Def� , and we

apply Theorem 14. Therefore there exists a �nitely generated subobject K � L

such that the induced homomorphism � : L=K ! �i2IPi=�(K) is a splitting

monomorphism. But Coker(�) �= M , hence M is isomorphic to a direct summand

of �i2IPi=�(K). Since �(K) is �nitely generated we can view �(K) as a subobject

of a subsum �i2JPi=�(K), were J is a �nite subset of I. Since �i2JPi is �nitely

presented, it follows that �i2JPi=�(K) is also �nitely presented, hence M is a

direct summand of a direct sum of �nitely presented objects. Then M is pure-

projective. �

The next observation allows us to prove that, in order to study the commuting

properties with respect to direct unions, it is enough to restrict to defect functors

associated to the homomorphisms which appear in the canonical decomposition of

�.


Proposition 17. Let � : L! P be an epimorphism. Then for every direct system

F = (Mi; vij)i;j2I of monomorphisms, the canonical map ��Fis a monomorphism.

Proof. Let x 2 Ker(��F). Using the notations from diagram (D1), there exists

y 2 lim�!Hom(L;Mi) such that x = lim�!�i(y) and LF(y) factorizes through �. There

exists i 2 I and �i : L! Mi such that y = vi(�i), where vi denotes the canonical

map vi : Hom(L;Mi)! lim�!Hom(L;Mi). Then �Fvi(�i) = vi�i factorizes through

�. Let : P ! lim�!Mi be a homomorphism such that vi�i = �.

Let K = Ker(�) and �K : K ! L be the canonical homomorphism. Then

vi�i�K = ��K = 0. Since the structural homomorphisms vi are monomorphisms

we obtain �i�K = 0, hence �i factorizes through �. Then x = 0, and the proof is

complete. �

Corollary 18. Let � : L! P be an epimorphism. The following are equivalent:

(1) the functor Def� commutes with direct unions;

(2) for every direct family F of monomorphisms the induced homomorphism ��F

is an epimorphism;

Using Theorem 3 and Proposition 17 we have the following result:

Corollary 19. Suppose that � : L! P is a homomorphism in the abelian category

C, iK : K ! L is the kernel of �, �K : L ! L=K is the canonical epimorphism,

and � : L=K ! P is the homomorphism induced by �. Then ��Fis an isomorphism

(epimorphism) for a direct family of monomorphisms F if and only if ��Fand ��K

F

are isomorphisms (epimorphisms).

Proof. In order to prove the equivalence, let us remark, using the fact that direct

limits are exact in C, that for every direct family F we have a commutative diagram

0 // lim�!Def�(Mi)

��F

��

// lim�!Def�(Mi)

��F

��

// lim�!Def�K (Mi)

��KF

��

// 0

0 // Def�(lim�!Mi) // Def�(lim�!Mi) // Def�K (lim�!Mi) // 0;

and ��KF

is monic by Proposition 17. Now the equivalence stated in this corollary

is obvious. �

The condition ��KF

is an epimorphism can be replaced by a factorization condi-

tion:

Lemma 20. Let � : L ! P be a homomorphism and F = (Mi; vij)i;j2I a direct

system. Consider the following statements:

(1) ��Fis an epimorphism;

(2) (a) if �K : K ! L is the kernel of �, then for every homomorphism f :

L! lim�!Mi there exists i 2 I and h : L! Mi such that f�K = vih�K

(i.e. the restriction of f to factorizes through the canonical map vi),


(b) if � : L=K ! P is induced by � then ��Fis an epimorphism.

Then (2)) (1). If all vi are monomorphisms we have (1), (2).

Proof. (2) ) (1). Let f : L ! lim�!Mi be a homomorphism. Using (a) we can �nd

i 2 I and h : L ! Mi such that f�K = vih�K . Then (f � vih)�K = 0, hence there

exists � : L=K ! lim�!Mi such that �� = f � vih.

Using (b) and Lemma 12, we can �nd j 2 I, g : P ! lim�!Mi and : L=K !Mj

such that � = g� + vj . We can suppose i = j. Then f � vih = g�� + vi �, hence

f = g� + vi( � + h). Another application of Lemma 12 completes the proof.

(1)) (2) Let f : L! lim�!Mi be a homomorphism. Using Lemma 12 there exist

h : L!Mi and g : P ! lim�!Mi which are homomorphisms such that f = g�+ vih.

Then f�K = vih�K , hence (a) is valid.

The condition (b) follows from Corollary 19. �

Remark 21. In fact the condition (b) in the above lemma can be proved directly.

In order to do this, let us consider a homomorphism f : L=K ! lim�!Mi. If � : L!

L=K is the canonical projection then we can �nd i 2 I and two homomorphisms

h : L ! Mi, g : P ! lim�!Mi such that f� = g� + vih. Then vih(K) = 0. Since vi

is a monomorphism, we have h(K) = 0. It follows that there exists h : L=K !Mi

such that h = h�: Then (g� + vih)� = f�, hence g� + vih = f , and the proof is

complete.

In fact the case when ��Fis an epimorphism for all direct systems of monomor-

phisms can be characterized in the following way:

Theorem 22. Let � : L! P be a homomorphism in C. The following are equiva-

lent:

(1) for every direct system F = (Mi; vij)i;j2I of monomorphisms ��F

is an

epimorphism;

(2) (a) if �K : K ! L is the kernel of �, then K can be embedded in a �nitely

generated subobject H � L, and

(b) if � : L=K ! P is induced by � then ��Fis an epimorphism for all

direct systems of monomorphisms F = (Mi; vij)i;j2I .

Proof. (1))(2) We apply Lemma 20 to obtain (b). For (a), we apply Theorem 14

to �nd a homomorphism g : P ! L such that Im(1L � g�) can be embedded in a

�nitely generated subobject H of L. Then K can be also embedded in H.

(2))(1) It is enough to prove that for every direct system of monomorphisms and

for every f : L! lim�!Mi there exists i 2 I and h : L!Mi such that f�K = vih�K .

Let f : L ! lim�!Mi be a homomorphism. By (a) there exists a factorization

�K = �H�KH . Since H is �nitely generated there exists an index i 2 I such that

iHf factorized through vi. Therefore there exists h : L!Mi such that f�H = vih,

hence f�k = vih�K . �


In the end of this section we come back to the general case, in order to char-

acterize the functor Def� associated to a monomorphism � : L ! P for the

case when we can �nd a subobject H � L such that the induced homomorphism

� : L=H ! P=�(H) is split mono.

Proposition 23. Let � : L ! P be a monomorphism and H a subobject of L.

If � : L=H ! P=�(H) is the homomorphism induced by � then we have an exact

sequence of functors

(P=�(H);�)��

! (L=H;�)! Def� ! Def��(H)! Def�H ! 0;

and the following are equivalent:

(1) the induced homomorphism � : L=H ! P=�(H) is a splitting monomor-

phism;

(2) the induced sequence of functors

0! Def� ! Def��(H)! Def�H ! 0

is exact.

Proof. We have a commutative diagram

0

��

0

��

0 // H

�H

��

�0// �(H)

��(H)

��

// 0

0 // L�

//

�H

��

P //

��(H)

��

M // 0

0 // L=H�//

��

P=�(H) //

��

M // 0

0 0


with exact sequences, which induces a the solid part of the following commutative

diagram of functors and natural transformations

0

��

0

��

0 // (M;�) // (P=�(H);�)

��(H)

��

��

// (L=H;�) //___

��H��

Def�(�)

0 // (M;�) // (P;�)

��(H)

��

��// (L;�)

��//

��H��

Def�(�) //

0

��

0

0 // (�(H);�)�0�

//

��(H)

��

(H;�)

��H��

// 0

Def��(H)(�) //

��

Def�H (�)

��

// 0

0 0

in which all lines and columns are exact sequences. Applying the snake lemma we

obtain the natural transformation (L=H;�) 99K Def�(�) such that the sequence

(P=�(H);�)��

! (L=H;�) 99K Def� ! Def��(H)! Def�H ! 0

is exact.

Now the equivalence (1),(2) is obvious since � is split mono i� the natural

homomorphisms HomC(�;X) are epimorphisms for all X 2 C.

�

5. Commuting with direct sums

Let F = (Mi)i2I be a family of objects in C and �i :Mi !L

iMi the canonical

monomorphisms. Recall that

��F:M

i

Def�(Mi)! Def�(M

i

Mi)

denotes the natural homomorphisms induced by the family Def�(�i), i 2 I. It

is easy to see that �MF is a monomorphism, since (MF )�1(Im(Hom(�;

LiMi) =L

i Im(Hom(�;Mi)). Moreover, Def� commutes with �nite direct sums (it is addi-

tive).

For every family of objects (Mi; i 2 I) and for J � I denote by �J the canonical

projectionL

i2I Mi !L

i2J Mi.

Theorem 24. If � : L! P is a homomorphism and (Mi; i 2 I) a family of objects,

the following are equivalent:

(1) Def� commutes with direct sum of (Mi; i 2 I).


(2) For every homomorphism f : L !L

i2I Mi there exist a �nite subset

F � I, and a homomorphism g : P !L

i2InF Mi such that �InF f = g�.

Proof. SinceL

i2I Mi is a direct limit of the system (L

i2F Mi; F 2 I<!) with

canonical inclusions, ��Fis a monomorphism. Now it remains to apply Lemma 12.

�

If I is a set, X � I, and Mi, i 2 I, is a family of objects, we denote by

�IX : Def�(

M

i2I

Mi)! Def�(M

i2X

Mi)

the canonical epimorphism which is induced by the canonical mapL

i2I Mi !Li2X Mi. Note that �

IX is a splitting epimorphism of abelian groups.

Using a standard set-theoretical argument under assumption (V = L) we prove

that commuting of the functor Def� with countable direct sums is equivalent to

commuting with arbitrary direct sums. First, we make an easy observation:

Lemma 25. Let Mi, i 2 I, be a family of objects. Then Def� commutes withLi2I Mi if and only if for every � 2 Def�(

Li2I Mi) there is a �nite subset F � I

such that �IInF (�) = 0.

A cardinal � = jIj is !-measurable if it is uncountable and there exists a

countably-additive, non-trivial, f0; 1g-valued measure � on the power set of I such

that �(I) = 1 and �(fxg) = 0 for all x 2 I. We recall that if such a cardinal exists

then there exists a smallest !-measurable cardinal � and all cardinals � � � are

also !-measurable.

Proposition 26. Let � be a cardinal less than the �rst !-measurable cardinal. If

Def� commutes with countable direct sums then Def� commutes with direct sums

of � objects.

Proof. Let Ki, i 2 I, be a family of modules such that I is of cardinality � and

� 2 Def�(L

i2I Ki) is a �xed extension. By Lemma 25 it is enough to prove that

there is a �nite subset F � I such that �IInF (�) = 0. Let consider the set

I(I) = fX � I j there is a �nite subset F � X such that �IXnF (�) = 0g:

Suppose that I =2 I(I). We claim that there exists a subset Y � I such that

I(Y ) is a non-principal !1-complete ideal. Let us observe that for every subsets

F � X � I we have �IXnF = �X

XnF�IX . Furthermore, it is not hard to see that

I(I) contains ; and it is closed with respect to subsets and �nite unions. In order

to complete the proof of our claim it is remains to prove that if Xn, n 2 N, is a

countable set of pairwise disjoint subsets of I then there exists n0 2 N such thatSn�n0

Xn 2 I(I).

Let Xn, n 2 N, be a family of pairwise disjoint subsets of I, and X =Sn2NXn.

Since Def� commutes with countable direct sums, the canonical homomorphismM

n2N

Def�(M

i2Xn

Ki)! Def�(M

n2N

(M

i2Xn

Ki)) = Def�(M

i2X

Ki)


is an isomorphism. Therefore there is a positive integer n0 such that

�ISn�n0

Xn(�) = �XS

n�n0Xn

(�IX(�)) = 0;

henceSn�n0

Xn 2 I(I).

Now we claim that there exists Y =2 I(I) such that for every subset Z � Y with

Z =2 I(I) we have Y n Z 2 I(I).

Suppose by contradiction that such a Y does not exists. It follows that for every

Y =2 I(I) there exists a nonempty subset Z � Y such that Z; Y n Z =2 I(I). Since

; 6= I =2 I(I) we can �nd a partition I = Z0 [ Y1 such that Z0; Y1 =2 I(I). Now

; 6= Y1 =2 I(I), hence there exists a partition Y1 = Z1 [ Y2 such that Z1; Y2 =2 I(I).

We continue inductively this kind of choice: if Yn is constructed then exists a

partition Yn = Zn [ Yn+1 such that Zn; Yn+1 =2 I(I). Therefore we obtain a

countable sequence of sets Zn =2 I(I), and it is not hard to see that these sets

are pairwise disjoint. But, by what we proved so far, there exists n0 such thatSn�n0

Zn 2 I(I), a contradiction.

Then there exists a subset Y � I such that Y =2 I(I) and for every subset

Z � Y with Z =2 I(I) we have Y n Z 2 I(I). It is easy to see that we can de�ne

an !-additive f0; 1g-valuated map � on the power-set of I via the rule �(U) = 1

if U \ Y =2 I(I), and �(U) = 0 otherwise. It follows that I is !-measurable, a

contradiction. �

Corollary 27. Assume (V = L). If Def�(�) commutes with countable direct sums

then Def�(�) commutes with all direct sums.

It is well-known that Hom(M;�) commutes with countable direct sums i� it

commutes with all direct sums. Furthermore, as a consequence of the previous

result and Example 2 we obtain

Corollary 28. Let (V = L) and M 2 C. Then Ext1C(M;�) commutes with count-

able direct sums if and only if Ext1C(M;�) commutes with all direct sums.

Remark 29. We don't know what it happens if we remove the set theoretic as-

sumption (V = L). For the case of abelian groups it can be proved, as in [4,

Theorem 5.3] that if M is an abelian group such that Ext1Ab(M;�) commutes with

countable direct sums then M is a Whitehead group. On the other side, the same

result shows us that if Ext1Ab(M;�) commutes with all direct sums then M is free.

The interested reader can �nd some similar phenomena in [3, Section 2].

The following results characterize when Def� commutes with countable direct

sums. It generalizes a classical characterization of small modules proved by

Rentschler in [29].

Proposition 30. Let � : L! P be a homomorphism. The functor Def� commutes

with countable direct sums if and only if for every countable chain of subobjects

(D) : L0 ,! L1 ,! L2 ,! : : :


such that L is a direct union of (D) there exists n for which the induced map

�0 : L=Ln ! P=�(Ln) is a splitting monomorphism.

Proof. ()) For each i 2 N denote by �i : Li ! L the canonical monomorphism

and put Ai = Coker(�i) �= L=Li. Suppose that � : L !L

iAi is the morphism

de�ned by direct sum of the canonical epimorphisms �i : L ! Ai, i.e. �i� = �i,

where �i :L

iAi ! Ai is the canonical projection. By the hypothesis, there

exist a �nite subset F � N and a homomorphism g : P !L

i2NnF Ai such that

�NnF� = g�. Let n =2 F . If �n : L! An represents the canonical epimorphism, we

obtain �n = �ng�, hence �ng�(Ln) = 0. Then �ng factorizes through the canonical

epimorphism �n : P ! P=�(Ln), so �ng = g0�n and g0 : P=�(Ln)! An.

We obtain g0�0�n = g0�n� = �ng� = �n, hence g0�0 = 1An , and the proof is

complete.

(() Let � 2 Hom(L;L

i<! Ai) and denote by ��n the canonical epimorphismsLiAi !

Li�nAi. Obviously, the family Ln = Coker(��n�) with canonical

monomorphisms forms an increasing chain such that L is its direct union.

By the hypothesis there exists n such that �0 : L=Ln ! P=�(Ln) has a left

inverse g0 : P=�(Ln)! L=Ln.

If we put F = f1; : : : ; ng and g : P !L

i>nAi, g = g0�n, where �n : P !

P=�(Ln) is the canonical epimorphism, then we can apply Theorem 24 to obtain

the conclusion. �

We will say that the homomorphism � : L ! P is �-splitting small, where �

is a cardinal, if for every system of objects (Ai; i < �) and for every homomor-

phism � : L !L

i<�Ai there exists a �nite subset F � � such that the cokernel

homomorphism � in the pushout diagram

(D2)

L�

��! P ��! U ��! 0

��nF�

??y??y

L

i2�nF Ai ��! X�

��! U ��! 0

splits.

Now, we make an elementary observation.

Lemma 31. Consider a commutative diagram with exact rows and columns

(D3)

A0�0��! B0

�0��! C ��! 0

�0

??y �0

??y

A1�1��! B1

�1��! C ��! 0

:

If �0 : B0=Ker�0 ! C ! 0 induced by (D3) splits, then �1 splits as well.

Proof. Let �0 : B0=Ker�0 ! B1 the homomorphism induced by �0. Since there

exists � : C ! B0=Ker�0 such that idC = �0� = �1�0�, the homomorphism �1

splits. �


Proposition 32. The homomorphism � is !-splitting small if and only if Def�

commutes with countable direct sums.

Proof. ()) We consider a countable chain of subobjects

(D) : L0 ,! L1 ,! L2 ,! : : :

such that L is a direct union of (D).

For each i denote by �i : Li ! L the canonical monomorphism and put Ai =

Coker(�i) �= L=Li. Suppose that � : L !L

iAi is the homomorphism de�ned by

direct sum of the canonical projections. By the hypothesis, there exists a �nite

subset F such that the homomorphism � in the pushout diagram (D2) splits. Let

n =2 F . Clearly, the homomorphism � in the pushout diagram

(D4)

Li2�nF Ai ��! X

��! U ��! 0

??y??y

An ��! Y�

��! U ��! 0

splits. Since �n is an epimorphism and the composition of the pushout diagrams

(D2) and (D4) is so, the diagram

L�

��! P ��! U ��! 0

�n

??y �

??y

An ��! Y�

��! U ��! 0

commutes and � is an epimorphism. As � splits, it remains to observe that Ker� =

�(Ln).

(() Let � 2 Hom(L;L

i<! Ai) and denote by ��n the canonical epimorphismsLiAi !

Li�nAi. Obviously, the family Ln = Coker(��n�) with canonical

monomorphisms forms an increasing chain such that L is its direct union.

By the hypothesis there exists n such that �(L)=�(Ln) is a direct summand of

P=�(Ln), hence it remains to put F = f1; : : : ; ng and to apply Lemma 31 on the

pushout diagram (D2). �

Now we will apply the above results in order to see when Ext1C(M;�) commutes

with direct sums.

Lemma 33. Let � be a cardinal, and consider an exact sequence L�! P !M ! 0.

Then � is �-splitting small if and only if Def� commutes with direct sums of �

objects.

Proof. ()) Let Ai; i � � be a family of objects. Let � 2 Def�(L

i<�Ai), and

consider a homomorphism � 2 HomC(L;L

i<�Ai) whose coset modulo the image


of Hom(�;L

i<�Ai) is �. We construct the pushout diagram

(D5)

L�

��! P�

��! M ��! 0

�

??y �

??y

Li<�Ai

��! X

��! M ��! 0:

As � is �-splitting small, there exists a �nite set F � � such that the second row

of the pushout diagram

(D6)

L�

��! P�

��! M ��! 0

��nF�

??y??y

L

i2�nF Ai ��! Y ��! M ��! 0

:

splits. Thus ��nF (�) = 0. Now the assertion follows from Lemma 25.

(() Fix � : L !L

i<�Ai. Then there exist an object X and homomorphisms

� and � such that (D5) forms a pushout diagram. Since Def� commutes with the

direct sums of family (Ai; i < �), Lemma 25 imply that there exists a �nite subset

F � � such that the second row of the pushout diagram

0 ��!L

i<�Ai�

��! X�

��! M ��! 0

��nF

??y??y

0 ��!L

i2�nF Ai ��! Y ��! M ��! 0

splits. Thus we get pushout diagram (D6) where the second row splits, so � is

�-splitting small. �

Theorem 34. Let � be a cardinal less than the �rst !-measurable cardinal. The

following are equivalent for a homomorphism L�! P in C:

(1) The functor Def� commutes with direct sums of � objects,

(2) � is �-splitting small,

(3) � is !-splitting small.

Proof. (1) , (2) is the assertion of Lemma 33, (2) ) (3) is trivial and (3) ) (1)

follows from Proposition 26 and Lemma 33. �

We can apply the last assertion to see when an Ext1-covariant functor preserves

direct sums.

Corollary 35. Suppose that C has projective strong generator which is a direct sum

of �nitely presented objects. Let � be a cardinal less than the �rst !-measurable

cardinal. The following are equivalent for a projective presentation 0! L�! P !

M ! 0 of M 2 C:

(1) The functor Ext1C(M;�) commutes with direct sums of � objects,

(2) � is �-splitting small,

(3) � is !-splitting small.


Corollary 36. Let (V = L), and suppose that C has projective strong generator

which is a direct sum of �nitely presented objects. If M 2 C then Ext1(M;�)

commutes with all direct sums if and only if there exists a projective presentation

0! L�! P !M ! 0 for M such that � is !-splitting small.

In these conditions for all projective presentations 0 ! L0�0

! P 0 ! M ! 0 and

for all cardinals � the homomorphism �0 is �-splitting small.

We close this section with an application of Proposition 9 to the study of the

covariant Ext1-functor.

Lemma 37. Let 0 ! L ! P ! M ! 0 be an exact sequence such that P is

projective. If L and M are small objects, then M and P are �nitely generated and

Ext1(M;�) commutes with direct sums.

Proof. Since the class of small objects is closed with respect to extensions, P is

small, hence �nitely generated. Note that a direct sumL

i<�Ai is precisely direct

union of the diagram F = (L

i2F Ai; �FGj F � G 2 �<!) where �FG are the

canonical inclusionsL

i2F Ai !L

i2GAi. As all homomorphisms F from the

diagram (D1) are isomorphisms, �MF is isomorphism as well. �

Applying Proposition 9(1) we obtain the following

Corollary 38. Suppose that C has projective strong generator which is a direct

sum of �nitely presented objects. Let M be a �nitely generated object, and let

0! L�! P !M ! 0 be a projective presentation such that P is �nitely generated.

Then Ext1C(M;�) commutes with direct sums if and only if for L is small.

6. The covariant Ext1-functor and direct unions

In this and the next sections C will be an abelian category with a projective

strong generator which is a direct sum of �nitely presented objects.

Let us �x an object M in C. We will apply the previous results to study the

commuting properties for the covariant functor Ext1C(M;�). In order to do this,

we �x a projective presentation

0! L�! P !M ! 0;

and we can apply the previous results for the functor Def� .

In order to simplify our presentation we will say, as in [10], that the object M is

an fg-1-object (respectively fp-1-object) if there is a projective resolution

(P) : � � � ! P2 ! P1�1! P0 !M ! 0

such that 1(P) = Im(�1) is �nitely generated (respectively, �nitely presented),

i.e. there is a projective resolution for M such that the �rst syzygy associated to

this resolution is �nitely generated (�nitely presented). The object M is an FPn-

object if it has a projective resolution such that Pi are �nitely presented for all

i 2 f0; : : : ; ng.


It is proved in [19, Lemma 3.1.6] that if M is an FP2 object then Ext1C(M;�)

commutes with direct limits. Using Proposition 9 and Example 2(1) for the kernel

of a projective presentation P ! M ! 0, it is easy to see that for the �nitely

generated objects this hypothesis is sharp. We recall that in our hypothesis every

�nitely generated projective object is �nitely presented.

Corollary 39. Let M be a �nitely generated object.

(1) Ext1C(M;�) commutes with direct unions if and only if M is �nitely pre-

sented.

(2) Ext1C(M;�) commutes with direct limits if and only if M is an FP2-object.

Inductively, using the dimension shifting formula we obtain a version, of [12,

Theorem 2] and [32, Theorem A]:

Corollary 40. The following are equivalent for an object M :

(1) M has a projective resolution

(P) : Pn ! � � � ! P2 ! P1�1! P0 !M ! 0

such that Pi are �nitely presented for all 0 � i � n;

(2) The functors ExtiC(M;�) commute with direct unions for all 0 � i � n.

Remark 41. Corollary 39 can be reformulated in the following way: the functor

HomC(M;�) commutes with direct limits if and only if the functors HomC(M;�)

and Ext1C(M;�) commute with direct unions. The proof presented here uses the

existence of the strong generator U which is a direct sum of �nitely presented

projective objects. It is an open question if this result is valid in more general

settings, e.g. for general Grothendieck categories without enough projectives. For

instance this equivalence is valid for the category of all Abelian p-groups (p is a �xed

prime), which is a Grothendieck category without non-trivial projective objects, as

a consequence of [31, Theorem 5.4].

In order to prove the main result of this section, we say that a covariant functor

F : C ! Ab is isomorphic to a direct summand of a functor G : C ! Ab if we can

�nd two natural transformations � : F ! G and � : G! F such that �� = 1F .

Lemma 42. Let F;G : C ! Ab be additive covariant functors such that F is

isomorphic to a direct summand of G. If F = (Mi)i2I is a direct family such that

the canonical homomorphism �G : lim�!G(Mi)! G(lim�!Mi) is monic (epic) then the

canonical homomorphism �F : lim�!F (Mi)! F (lim�!Mi) is monic (epic).

Proof. If � : F ! G and � : G! F are natural transformations such that �� = 1F ,

we have the commutative diagram

0 ��! lim�!F (Mi)lim�!

�Mi��! lim�!G(Mi)

lim�!

�Mi��! lim�!F (Mi) ��! 0

�F

??y �G

??y �F

??y

0 ��! F (lim�!Mi)�lim�!

Mi

��! G(lim�!Mi)�lim�!

Mi

��! F (lim�!Mi) ��! 0

;


and the conclusion is now obvious. �

Corollary 43. Let M be an object such that Ext1C(M;�) commutes with the colimit

of a direct system F. Then every direct summand N of M has the same property.

Now we are ready to characterize when the covariant Ext1C functors commute

with direct unions. We recall thatM is 2-almost projective if it is a direct summand

of a direct sum P �F with P a projective object and F a �nitely presented object,

[15]. For reader's convenience we include a proof for the following characterization.

Theorem 44. The following are equivalent for an object M in C:

(1) The functor Ext1C(M;�) commutes with direct unions;

(2) M is a direct summand of an fg-1-object.

(3) M is a 2-almost projective object.

Proof. (1) ) (2) We consider a projective resolution 0 ! L�! P ! M ! 0. By

Theorem 14 there exists a �nitely generated object H � L such that the induced

homomorphism � : L=H ! P=�(H) is split mono. Since Coker(�) �= Coker(�) �=

M , it follows thatM is isomorphic to a direct summand of the fg-1-object P=�(H).

(2))(3) It is enough to assume thatM is an fg-1-object. IfM is such an object

then we can consider the diagram (D1) with P projective and L �nitely generated.

If U is an object such that P �U is a direct sum of copies of some objects from U ,

we consider the induced exact sequence 0 ! L�! P � U

��1U�! M � U ! 0. Let

P � U = �i2IPi, where all objects Pi are �nitely presented and projective. Since

L is �nitely generated, there is a �nite subset J � I such that �(L) � �i2JPi.

Therefore M � U �= (�i2InJPi) � (�i2JPi)=�(L) is a direct sum of a projective

object and a �nitely presented object.

(3))(1) In view of Corollary 43, we can assume that M is �nitely presented.

Then we apply Corollary 39. �

For arbitrarily direct limits, we have not a general answer. However, for some

particular cases, including coherent categories (C is coherent if every �nitely gen-

erated subobject of a projective object is �nitely presented), we can apply the

previous result. In order to do this, let us state the following

Proposition 45. Let M be an fg-1-object. The following are equivalent:

(1) M is an fp-1-object;

(2) Ext1C(M;�) commutes with direct limits.

Proof. (1))(2) Suppose that M is an fp-1-object. As in the proof for (2))(3)

in the previous theorem, we observe that there is a projective object L such that

M �L is a direct sum of an FP2-object and a projective object. Therefore we can

suppose that M is FP2. For this case the result is well known (see [19, Lemma

3.1.6]).


(2))(1) Let 0! K ! P !M ! 0 be an exact sequence such that K is �nitely

generated. Using again the proof of (2) ) (3) in the previous theorem, there is

a projective object K such that M � K = N � U , N = P 0=K, where P 0 is a

�nitely generated projective object and U is projective such that P 0 �U = P �K.

Then Ext1C(N;�)�= Ext1C(M;�) commutes with direct limits. Since P 0 is �nitely

presented we can use Lemma 9, and we conclude that K is �nitely presented. �

From this proposition and its proof we obtain some useful corollaries. First of

them allows us to construct examples of objectsM such that Ext1C(M;�) commutes

with respect to direct unions, but it does not commute with direct limits.

Corollary 46. Suppose that M is an fp-1-object and 0 ! L ! P ! M ! 0 is

an exact sequence such that P is �nitely generated projective. Then L is �nitely

presented.

Consequently, if for every �nitely presented object M the functor Ext1C(M;�)

commutes with direct limits then C is coherent.

Proof. We consider an exact sequence 0 ! L1 ! P1 ! M ! 0 such that P1 is

�nitely presented projective and L1 is �nitely presented. By Schanuel's lemma we

have L1 � P �= L� P1, and now the conclusion is obvious. �

In fact, for coherent categories (in particular for modules over coherent rings

or for the category of modules over the category mod-R) the functor Ext1C(M;�)

commutes with direct limits if and only it it commutes with respect direct unions:

Corollary 47. Suppose that M has a projective resolution (P) such that 1(P) is

a direct union of �nitely presented subobjects. The following are equivalent:

(1) Ext1C(M;�) commutes with direct limits;

(2) Ext1C(M;�) commutes with direct unions;

(3) M is a direct summand of an fp-1-object.

Proof. In the proof of Theorem 44 we can choose F such that all Mi are �nitely

presented. �

As each countably generated object is a direct union of a chain of �nitely gen-

erated modules, Theorem 44, Example 2(1) and the previous result implies the

following consequence:

Corollary 48. Let 0 ! L ! P ! M ! 0 be a short exact sequence with L

countably generated. If Ext1(M;�) commutes with direct sums then M is 2-almost

projective, hence Ext1(M;�) commutes with direct unions.

Moreover, for coherent categories we obtain from Theorem 44 a generalization

of [13, Theorem A]:

Theorem 49. Suppose that C is a coherent category. The following are equivalent

for an object M and a positive integer n:


(1) ExtnC(M;�) commutes with direct limits (unions);

(2) if m � n is an integer then then ExtmC (M;�) commutes with direct limits

(unions).

Proof. (1))(2) By dimension shifting formula we can assume n = 1, and it is

enough to prove that Ext2C(M;�) commutes with direct unions.

Let 0 ! L�! P ! M ! 0 be an exact sequence such that P is projective.

By Theorem 14(6) there exists a �nitely generated subobject H � L such that the

induced homomorphism �0 : L=H ! P=�(H) is split mono. Using Theorem 44 we

obtain that Ext1C(L=H;�) commutes with direct limits. Moreover, we can view H

as a �nitely generated subobject of P , hence H is �nitely presented. Therefore for

every directed family F = (Mij ; vij), in the commutative diagram

lim�!(H;Mi) //

HF��

lim�!Ext1C(L=H;Mi) //

�L=HF

��

lim�!Ext1C(L;Mi)

�LF��

// lim�!Ext1C(H;Mi)

�HF��

(H; lim�!Mi) // Ext1C(L=H; lim�!Mi) // Ext1C(L; lim�!Mi) // Ext1C(H; lim�!Mi)

the homomorphisms HF , �

L=HF

and �HF are isomorphisms. Then �LF is also an

isomorphism, and the proof is complete.

(2))(1) is obvious. �

7. Ext-steady rings

Recall that C means again an abelian category with a projective strong generator

which is a direct sum of �nitely presented objects and denote by U the corresponding

set of �nitely presented objects. We say that the category C is �nite ext-steady if for

every �nitely generated object M such that Ext1(M;�) commutes with all direct

sums it holds that Ext1(M;�) commutes with all direct unions.

Proposition 50. The following conditions are equivalent:

(1) C is �nite ext-steady,

(2) every small subobject of every projective object is �nitely generated,

(3) every small subobject of every �nitely generated projective object is �nitely

generated.

Proof. (1))(2) Let L be a small subobject of a projective object P , i.e. there

exist a cardinal �, a family Pi 2 U , i < �, and a monomorphism L !L

i<� Pi.

Moreover, as L is small, there exists F � � and a monomorphism � : L!L

i2F Pi.

Put M = Coker(�). By Lemma 37, the functor Ext1(M;�) commutes with direct

sums, hence it commutes with direct unions by the hypothesis. Thus L is �nitely

generated by Theorem 44.

(2))(3) Clear.

(3))(1) Let M be a �nitely generated object such that Ext1(M;�) commutes

with direct sums. Then homomorphisms MF , P

F and �MF from the diagram (D1)


are isomorphisms, �LF is isomorphism as well, hence L is small. By the hypothesis

L is �nitely generated. Thus Ext1(M;�) commutes with direct unions by Theo-

rem 44(4). �

We say that a unital ring R is right �nite ext-steady if the category of all right

R-modules is �nite ext-steady.

Example 51. It is proved in [33] that every in�nite product of unital rings contains

an in�nitely generated small ideal, hence it is not �nite ext-steady.

As every �nitely generated projective module is a direct summand of a �nitely

generated free module we obtain a consequence of the last proposition:

Corollary 52. A ring R is a right �nite ext-steady if and only if every small right

ideal is �nitely generated.

Proof. By Proposition 50 it is enough to prove that every small submodule I of

every �nitely generated free module Rn is �nitely generated. Proceed by induction,

the claim is clear for n = 1 hence suppose that n > 1 and denote by � : Rn !

R the canonical projection and by � : R ! Rn the canonical injection on the

�rst coordinate. As �(I) is small so �nitely generated submodule of R, and the

small module I + �(R)=�(R) �= I=(I \ �(R)) is embeddable into Rn�1, the module

I=(I \ �(R)) is �nitely generated by the induction hypothesis. Moreover, I=��(I)

and ��(I) are �nitely generated as well because I\�(R) � ��(I), hence I is �nitely

generated. �

Recall that a ring is right steady provided every small right module is necessarily

�nitely generated, [16]. However general structural ring-theoretical characterization

of right steady rings is still an open problem, various classes of rings are known

to be right steady (noetherian and perfect [29], semiartinian of countable Loewy

chain [16]). Let us remark here that the criterion of steadiness for commutative

semiartinian rings [30] and for regular semiartinian rings with primitive factors

artinian [36] has a similar form as Corollary 52 since steadiness is in this cases

equivalent to the condition that every small ideal of every factor-ring is �nitely

generated.

Corollary 53. The category of right modules over right steady or countable ring

is right �nite ext-steady.

Since there are known countable non-steady rings, as it is illustrated in the

following example, the inclusion of classes of steady rings and �nite ext-steady

rings is strict.

Example 54. Let F be a countable �eld and X is a in�nite countable set. Then

it is is well known that over the polynomial ring F hXi in noncommuting variables

X every injective module is small. Thus F hXi is a non-steady countable ring.


Remark 55. From Proposition 50 and [35, Example 14] we deduce that the ext-

steadiness property is not left-right symmetric.

We conclude the section with generalization of Corollary 48 in the case of modules

over perfect rings.

Proposition 56. Let R be a right perfect ring and M a right R-module such that

Ext1(M;�) commutes with direct sums. Then M is fg-1.

Proof. Let 0 ! L ! P ! M ! 0 be a projective presentation for M . Denote

by J the Jacobson radical of R. Since R is right perfect, L=LJ is semisimple,

hence L=LJ �=L

i2I Si for a family of simple modules (Si; i 2 I). We consider

f : L !L

i2I Si as the canonical projection. Then by Theorem 24 there exists a

�nite set F � I, and g : P !L

i2InF Si such that �InF f = g�.

Let � : Q!L

i2InF Si be a projective cover ofL

i2InF Si which exists because

R is right perfect. As � is surjective, there exists a homomorphisms � : P ! Q such

that �� = g. Note that Ker� = QJ is super uous in Q and �� = g� = �InF f , thus

��(L) = Q where Q is projective. Clearly, there exists a homomorphism ' : Q! P

such that ��' = idQ, hence L = '(Q)�Ker�� and P = �'(Q)�Ker� . Since the

factorization by '(Q) induces a short exact sequence

0! L='(Q)! P=�'(Q)!M ! 0

and P=�'(Q) is projective, it remains to prove that L='(Q) is �nitely generated.

This follows from the observations that V = Ker�� = L='(Q) and V=V J �=Li2F Si. �

Finally we summarize the results about connections between possible commuting

properties of a functor Ext1(M;�):

Corollary 57. For a right R-module M we consider the following possible proper-

ties:

(DS) Ext1(M;�) commutes with direct sums;

(DU) Ext1(M;�) commutes with direct unions;

(DL) Ext1(M;�) commutes with direct limits.

Then the following are true:

(1) If R is hereditary then (DS),(DU),(DL).

(2) R is right coherent if and only if (DU),(DL) for all right R-modules M .

(3) If R is right perfect then (DS),(DU).

Proof. (1) is a consequence of Proposition 11 (this is also proved in [32]).

(2) is proved in Theorem 49 and Corollary 46.

(3) is a consequence of Proposition 56. �


References

[1] J. Ad�amek, J. Rosick�y: Locally presentable categories and accessible categories, LondonMath. Soc. Lec. Note Series 189 (1994).

[2] J. Ad�amek, J. Rosick�y, E. Vitale, Algebraic Theories: A Categorical Introduction to GeneralAlgebra, Cambridge Tracts in Mathematics 184, Cambridge University Press, (2010).

[3] U. Albrecht, S. Breaz, P. Schultz: The Ext functor and self-sums, Forum Math. 26 (2014),851{862.

[4] U. Albrecht, S. Breaz, P. Schultz: Functorial properties of Hom and Ext, ContemporaryMathematics 576 (2012), 1{15.

[5] L. Angeleri-H�ugel, Frederik Marks, Jorge Vit�oria: Silting modules, arXiv:1405.2531.[6] M. Auslander: Coherent functors, Proc. Conf. Categor. Algebra, La Jolla 1965, (1966), 189{

231.[7] M. Auslander, I. Reiten, S.O. Smal�o: Representation theory of Artin algebras, Cambridge

Studies in Advanced Mathematics 36. Cambridge: Cambridge University Press, 1995.[8] S. Bazzoni, J. �S�tov��cek: Flat Mittag-Le�er modules over countable rings, Proc. Amer. Math.

Soc. 140 (2012), 1527{1533.[9] R. Bieri, B. Eckmann: Finiteness properties of duality groups, Commentarii Math. Helvet.

49 (1974), 74{83.[10] S. Breaz: Modules M such that Ext1R(M;�) commutes with direct limits, Algebras and

Representation Theory, DOI: 10.1007/s10468-012-9382-y.[11] S. Breaz: When Ext1(M;�) commutes with direct unions, unpublished manuscript.[12] K. S. Brown: Homological criteria for �niteness, Commentarii Math. Helvet. 50 (1975), 129{

135.[13] J. Cornick, I. Emmanouil, P. Kropholler, O. Talelli: Finiteness conditions in the stable module

category, Advances in Mathematics, 260 (2014), 375{400.[14] W. Crawley-Boevey: In�nite-dimensional modules in the representation theory of �nite-

dimensional algebras, Canadian Math. Soc. Conf. Proc., 23 (1998), 29{54.[15] V. Drinfeld: In�nite-dimensional vector bundles in algebraic geometry: an introduction, The

unity of mathematics, Progr. Math. 244 (2006), Birkh�auser, Boston, MA, 263-304.[16] P.C. Eklof, K.R. Goodearl and J. Trlifaj: Dually slender modules and steady rings, Forum

Math. 9 (1997), 61{74.[17] S. Estrada, P. Guil Asensio, M. Prest, J. Trlifaj: Model category structures arising from

Drinfeld vector bundles, Adv. Math. 231 (2012), 1417{1438.[18] S. Estrada, P. Guil Asensio, J. Trlifaj: Descent of restricted at MittagLe�er modules and

generalized vector bundles, preprint.[19] R. G�obel, J. Trlifaj: Endomorphism Algebras and Approximations of objects, Expositions in

Mathematics 41, Walter de Gruyter Verlag, Berlin (2006).[20] R. Hartshorne: Coherent functors, Adv. Math. 140 (1998), 44{94.[21] M. H�ebert: What is a �nitely related object, categorically?, Applied Categorical Structures,

21 (2013), 1{14.[22] C. U. Jensen, H. Lenzing: Model theoretic algebra: with particular emphasis on �elds, rings,

modules, Algebra, Logic and Applications 2, Gordon and Breach Science Publishers, NewYork, (1989).

[23] H. Krause: A short proof for Auslander's defect formula, Linear Algebra Appl. 365 (2003),267{270.

[24] H. Krause: The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), no. 707.[25] H. Krause: Functors on locally �nitely presented additive categories, Colloq. Math., 75 (1998),

105{132 .[26] H. Lenzing: Endlich pr�asentierbare Moduln, Arch. Math. (Basel) 20 (1969), 262{266.[27] G. Puninski: Pure projective modules over an exceptional uniserial ring, St. Petersburg Math.

J. 13(6) (2002), 175{192.[28] M. Raynaud, L. Gruson: Crit�eres de platitude et de projectivit�e, Inventiones Math. 13 (1971),

1-89.[29] R. Rentschler: Sur les objects M tels que Hom(M;�) commute avec les sommes directes, C.

R. Acad. Sci. Paris Sr. A-B 268 (1969), 930{933.[30] P.R�u�zi�cka , J. Trlifaj and J. �Zemli�cka: Criteria of steadiness, Abelian Groups, Module Theory,

and Topology, New York 1998, Marcel Dekker, 359{372.


[31] P. Schultz: Commuting Properties of Ext, Journal of the Australian Mathematical Society,to appear.

[32] R. Strebel: A homological �niteness criterion, Math. Z. 151 (1976), 263{275.[33] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents, in Abelian groups

and objects (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995, 467{473.[34] C. E. Watts: Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc.

11 (1960), 5{8.[35] J. �Zemli�cka, J. Trlifaj: Steady ideals and rings, Rend. Semin. Mat. Univ. Padova 98 (1997),

161-172.[36] J. �Zemli�cka: Steadiness of regular semiartinian rings with primitive factors artinian,

J.Algebra, 304 (2006), 500{509.

Babes�-Bolyai University, Faculty of Mathematics and Computer Science, Str. Mihail

Kog�alniceanu 1, 400084 Cluj-Napoca, Romania



Physics, Sokolovsk�a 83, 186 75 Praha 8, Czech Republic


Chapter 5

Re ection of categorical

properties to a ring structure

This chapter summarizes results on four classes of rings with similar correspon-dence between ring structure and categories of modules. The work was originallypublished as the following four papers:

F. Jan �Zemli�cka, Socle chains of abelian regular semiartinian rings, J. PureAppl. Algebra 217/6 (2013), 1018{1025.

G. Tom�a�s Penk, Jan �Zemli�cka, Commutative tall rings, J. Algebra Appl., 13/4(2014).

H. M. Tamer Kosan, Jan �Zemli�cka, Mod-retractable rings, Commun. Algebra42/3 (2014) 998{1010.

I. M. Tamer Kosan, Jan �Zemli�cka, On modules and rings with restricted min-imum condition, Colloq. Math., 140,1 (2015), 75{86.

93

F. SOCLE CHAINS OF ABELIAN REGULAR SEMIARTINIAN

RINGS

JAN �ZEMLI�CKA

Abstract. Let R be an abelian regular and semiartinian ring with socle chain(S� j � � �). If �� denotes the rank of the semisimple module S�+1=S�, forevery � < �, then the dimension sequence (�� j � < �) is an invariant for R.By applying classical results of combinatorial set theory we prove necessaryconditions satis�ed by this invariant. On the other hand, we present construc-tions of commutative regular semiartinian rings with given ranks of slices ofsocle chain. In some particular cases we prove a necessary and su�cient con-dition under which there exists an abelian regular semiartinian ring with givenranks of slices.

Given a right moduleM over some ring R, the socle chain ofM is the increasing

chain of submodules (S� j � � 0) de�ned by the following rules: set S0 = 0 and,

recursively, S�+1=S� = Soc(M=S�) (we denote by Soc(M) the socle of M) for each

ordinal � and S� =S�<� S� if � is a limit ordinal. The �rst ordinal � such that

S� = S�+1 is called the socle length of M and one says that M is semiartinian if

S�(M) = M . This notion was examined �rst by L�aszl�o Fuchs in [8], however the

idea to study it in the particular case of ideals in commutative noetherian rings

goes back to Wolfgang Krull [12].

A ring R is said to be right semiartinian provided it is semiartinian as a right

R-module. It is easy to see that every right artinian module as well as every right

module over a right artinian (semiartinian) ring is semiartinian. Moreover, the

classical Bass's result [3, Theorem P, (1)!(7)] asserts that every non-zero right

module over a left perfect ring has a non-zero socle, hence every left perfect ring is

right semiartinian. Basic structural results about semiartinian rings were presented

by Constantin N�ast�asescu and Nicolae Popescu in [13]. They proved that a ring

R is right semiartinian if and only if its Jacobson radical J(R) is left T -nilpotent

and R=J(R) is right semiartinian [13, Proposition 3.2]. The works [1, 2, 5, 14, 18]

are focused on the important case of semiartinian rings whose Jacobson radical is

zero, note that such rings are von Neumann regular if they are commutative by [13,

Th�eor�eme 3.1]. Various constructions of semiartinian rings and modules of given

socle lengths are presented in [4, 5, 6, 8, 15] and properties of semiartinian rings

close to commutative has been studied in [1, 14, 18].

2000 Mathematics Subject Classi�cation. Primary 16E50; Secondary 16D60, 16S50.Key words and phrases. semiartinian ring, von Neumann regular ring, dimension sequence.This work is part of the research project MSM 0021620839, �nanced by M�SMT..

94

F. SOCLE CHAINS OF ABELIAN REGULAR SEMIARTINIAN RINGS 95

The dimension sequence of a regular semiartinian ring whose all primitive factors

are artinian, which re ects the structure of single semisimple slices, was introduced

in [14]. Dimension sequences appear to be a useful tool for investigation of the

global structure of these rings and of corresponding module categories, however they

do not completely describe categorical properties of classes of all modules. Pairs

of examples of semiartinian rings R�, T� with non-equivalent module categories

which have the same dimension sequences are presented in [6] for every non-limit

uncountable socle length �. The class of all modules over R� contains an in�nitely

generated small module while the ring T� does not satisfy this property.

The present paper is motivated by the question of which sequences can be rep-

resented as the dimension sequence of some abelian regular semiartinian ring, or

more precisely, which ranks of semisimple socle slices are possible. Proposition 3.1

gives a �rst estimation of relationships between ranks of slices of an abelian reg-

ular semiartinian ring. Theorem 3.5, which is based on results of combinatorial

set theory, enhances Proposition 3.1 under the Generalized Continuum Hypoth-

esis. Speci�cally, let R be an abelian regular, semiartinian ring with dimension

sequence (�� j � < �). Then we prove that ��+� � �� if cf(��) > max(j�j; !)

and ��+� � �+� otherwise. On the other hand, if a family of cardinals ��, � � �,

satis�es conditions (a) �� +� whenever cf(��) = !, and �� otherwise;

(b) jf� j � � � � �gj � ��, (c) �� < ! i� � = �, we construct examples of

commutative regular semiartinian rings with the rank of the �-slice of the socle

chain equal to �� (Theorem 5.1). Note that our constructions are inspired by the

works [4] and [6]. Criteria on dimension sequences in particular cases are given in

Theorem 5.2 and Corollary 5.3. Examples 4.7 and 5.4 illustrate boundaries of our

tools and dependency of the results on set theoretical axioms.

1. Preliminary

Throughout the paper a ring (K-algebra) R means an associative unitary ring

(K-algebra) and a module M means a right R-module. Recall that R is regular

if for each x 2 R there exists y 2 R satisfying xyx = x and R is abelian regular

provided it is a regular ring whose all idempotents are central. It is well known

that every �nitely generated (right) ideal of an abelian regular ring is generated

by some central idempotent. For further properties of abelian regular and regular

rings we refer to the monograph [10].

The socle ofM , i.e. the sum of all simple submodules ofM , is denoted by Soc(M)

and M is said to be semisimple provided that M = Soc(M). If M is semiartinian

with the socle chain (S�j � � �), the ordinal � is called the socle length of M

and the semisimple module S�+1=S� is the �-th slice of M in this paper. Note

that every �nitely generated semiartinian module is of a non-limit socle length and

every module over a right semiartinian ring of socle length � is semiartinian of socle

length less or equal to �. The least cardinality of any set of generators of M is


denoted by gen(M). If M is semisimple and in�nitely generated, gen(M) will be

called rank of M , and the rank of a �nitely generated semisimple module is de�ned

as its uniform dimension. We refer to [9] for a survey and generalizations of the

theory of semiartinian modules.

Let X be a set and � a cardinal. Cardinality of X is denoted by jXj, and the

symbol �+ means the successor of �, i.e. the least cardinal greater then �. Note

that we identify cardinals with the least ordinals of given cardinality, in particular,

! means the �rst in�nite cardinal (ordinal) and the �rst uncountable cardinal (or-

dinal) is denoted by !1. The symbol [X]� ([X]<�) means the set of all subsets of

X of cardinality � (less than �), and h�; �i stands for the interval f j � � � �g

for every ordinals � � �. As our results depend on a model of the set theory, note

that we work in Zermelo-Fraenkel system with the axiom of choice. For further

properties of set theoretical notions including elementary cardinal arithmetics we

refer to [11].

We recall a natural description of slices of regular semiartinian rings with prim-

itive factors artinian.

Theorem 1.1. [14, Theorem 2.1] Let R be a right semiartinian ring and let L =

(S� j � � � + 1) be the right socle chain of R. Then the following conditions are

equivalent:

(1) R is regular and all right primitive factor rings of R are right artinian,

(2) for each � � � there are a cardinal ��, a positive integer n�� and a skew

�eld K�� for each � < �� such that S�+1=S� �=L

�<��Mn�� (K��), as

rings without unit. The pre-image of Mn�� (K��) coincides with the �-

th homogeneous component of R=S� and it is �nitely generated as a right

R=S�-module for all � < ��. Moreover, �� is in�nite if and only if � < �.

If (1) holds true, then R is also left semiartinian, and L is the left socle chain of R.

In general, for any regular ring the condition of being semiartinian is right/left

symmetrical, as the right and left socle chains coincide (see [1, Proposition 2.3]).

Suppose that R is a regular semiartinian ring of socle length � + 1 such that all

right primitive factors of R are right artinian. Following [14] we de�ne

D(R) = f(��; f(n�� ;K��) j � < ��g) j � � �g

as the dimension sequence of R; it collects invariants determined by Theorem 1.1.

The structural question about relationships between single slices S�+1=S� may be

formulated in the language of Theorem 1.1. Namely, we ask (and partially answer

in this paper) which sequences are representable as the dimension sequence of some

regular semiartinian rings with primitive factors artinian.

Let � < �. Since each homogeneous component of R=S� is injective by [10,

Proposition 6.18], it is a ring direct summand of R=S�. Thus Theorem 1.1 induces

a surjective ring-homomorphism �� : R=S� ! Mn�� (K��) for each � < �� and

the family (��)�<�� induces the canonical map '� : R=S� !Q�<��

Mn�� (K��).


Finally, denote by E(R) the set of all central idempotents of R and put E(I) =

I \ E(R) whenever I � R.

First, we prove basic observations concerning dimension sequences and central

idempotents.

Lemma 1.2. Let � be a cardinal, R a ring for every < �, and S a subring ofQ <�R containing

L <�R . Suppose that e 2 S. Then e is a central idempotent

of S i� e is a central idempotent ofQ <�R .

Proof. It is enough to prove that each central element e of S commutes with all

elements ofQ <�R , which follows immediately from the fact that e commutes

with every element of every ring summand R �Q <�R , < �. �

Lemma 1.3. Suppose that R is a regular and right semiartinian ring with all

primitive factors right artinian, let (S� j � � � + 1) be its socle chain and

f(��; f(n�� ;K��) j � < ��g) j � � �g its dimension sequence. Then the following

hold:

(1) For every � � �, the canonical map '� induces an isomorphism from R=S�

onto a subring ofQ�<��

Mn�� (K��) which containsL

�<��Mn�� (K��).

(2) jE(R)j � 2�0 .

Proof. (1) Let � � �. Since the socle of R=S� is isomorphic as a ring without unit

toL

�<��Mn�� (K��) by Theorem 1.1 and since R=S� is a semiartinian ring, we

get that Soc(R=S�) \Ker('�) = 0. Hence '� is an injective homomorphism.

(2) Using the notation established above, �0�(e) is a central idempotent of

Mn0� (K0�) for any central idempotent e 2 R, hence �0�(e) is either the zero

or unit matrix. Applying (1) we get that e is a central idempotent i� '0(e) is

a central idempotent of S =Q�<�0

Mn0� (K0�). Now, jE(R)j � jE(S)j = 2�0 by

Lemma 1.2. �

Lemma 1.4. Let R be a regular semiartinian ring whose all right primitive factors

are right artinian. Suppose that R is not semisimple, I � Soc(R) is an ideal, and

e 2 E(R).

(1) E(I) is �nite , I = fR for some f 2 E(I),

(2) if E(I) is in�nite and e 2 E(I), then jE(I)j = gen(I) = jE((1� e)I)j,

(3) E(eR) is �nite , E(eSoc(R)) is �nite , e 2 Soc(R).

Proof. (1) is a direct consequence of [10, Corollary 6.8], while (2) and (3) are

straightforward consequences of Lemma 1.3 and the fact that I is a direct sum of

homogeneous components of Soc(R), as it follows from [2, Corollary 1.12] �

We will use freely the following consequence of Lemma 1.4(2).

Corollary 1.5. Let R be a regular semiartinian ring whose all right primitive fac-

tors are right artinian. If f(��; f(n�� ;K��) j � < ��g) j � � �g is the dimension


sequence and (S� j � � � + 1) is the socle chain of R, then �� = gen(S�+1=S�) =

jE(S�+1=S�)j for every � < �.

It is well-known that the class of all semiartinian modules is closed under sums,

submodules, factors and extensions and that every module over a semiartinian ring

is semiartinian. We conclude this section by the several technical observations about

socle chains of direct sums of semiartinian modules and rings, which we will use

frequently in our constructions. The �rst two are easy and the third is a corrected

version of [1, Proposition 4.7].

Lemma 1.6. Let � be a cardinal and (M j < �) a family of semiartinian modules.

Denote by � the socle length of M and by (M� j � � � ) the socle chain of M

for every < �. Then (L

<�M� j � � �) is the socle chain of M =L

<�M ,

where � = sup <� � and M� =M for every < � and every � > � .

Corollary 1.7. Let n be a natural number and f(��; f(1;K��) j � < ��g) j � � �g

the dimension sequence of an abelian regular semiartinian ring R. Then Rn is an

abelian regular semiartinian ring of socle length �+1. Moreover, if f(�0�; f(1;K0��) j

� < �0�g) j � � �g is the dimension sequence of Rn, then �0� = n � �� and �0� = ��

for each � < �.

Proposition 1.8. Let � be an in�nite cardinal, K a �eld and R a K-algebra with

socle length � and socle chain (S� j � � � ) for each ordinal < �. Let us

consider the K-subalgebra R =L

<�R +K of the direct productQ <�R and

put � = sup <� � . If either � is limit or the set f j � = �g is in�nite, then:

(1) For every ordinal � � �, the �-th member of the socle chain of R isL

<� S� .

(2) If each R is right semiartinian, then R is right semiartinian with socle

length � + 1.

(3) R is regular, unit regular, directly �nite, has right primitive factor rings ar-

tinian or is a right V -ring if and only if R have the corresponding property

for all < �.

Proof. The proof works as in [1, Proposition 4.7] where only the argument K 6�L

<� S� is missing. Clearly, if all R are right semiartinian, R is right semiar-

tinian of a non-limit socle length equal either to � or � + 1. Thus the socle length

of R is � + 1 whenever � is limit. If f j � = �g is an in�nite set, then � = �+ 1

is a non-limit ordinal and the socle of R=L

<� S� is in�nitely generated, hence

R 6=L

<� S� . �

2. Some combinatorial set theory

Before we apply tools of combinatorial set theory to families of idempotents

generating single slices of abelian regular semiartinian rings, we recall basic notation


and several corresponding results. Note that we follow formulations from [17], which

is more convenient for our purpose than the original version of the results.

Recall that a subset A of a well-ordered set (X;�) is co�nal if for every x 2 X

there exists a 2 A such that x � a. The co�nality of an ordinal (cardinal) � is the

least cardinality of a co�nal subset of (�;�), we denote this cardinal by cf(�). Let

X be a set and � be a cardinal. A family S of subsets of X is called almost disjoint

with degree of disjunction �, provided jA\Bj < � for every di�erent sets A;B 2 S.

Finally, recall that the Generalized Continuum Hypothesis (GCH) is the statement

2� = �+ for every in�nite cardinal � [17, p.x], [11, p.55].

The following classical combinatorial result due to Tarski [16] plays an impor-

tant role for improving the necessary condition which will be expressed in Proposi-

tion 3.1(4).

Theorem 2.1. [17, Theorem 1.1.6] Suppose that GCH holds. Let � and � be

cardinals such that cf(�) < cf(�). If S is an almost disjoint family of subsets of �

such that jAj � � for every A 2 S and degree of disjunction of S is at most �, then

jSj � �.

Suppose X is a set, �, , � cardinals and m a natural number. Let us call a

decomposition of X a collection of (possibly empty) subsets of X with union X. A

decomposition � = f��j � < g is a partition provided �� \�� = ; and �� 6= ;

for every � < � < . The ordinary partition symbol �! (�)m means that for every

set X with jXj = � and for every decomposition � = f��j � < g of [X]m into

parts, there exist an ordinal � < and a subset H of X such that jHj = � and

[H]m � ��.

As the original results about ordinary partition symbol proved by Erd}os and

Rado in [7] are formulated for partitions, we recall an easy observation [17, p. 25]:

Lemma 2.2. Let �, , � be in�nite cardinals, m a natural number, and � �.

Then �! (�)m i� for every partition � = f��j � < g of [�]m into parts, there

exist an ordinal � < and a subset H of � such that jHj = � and [H]m � ��.

Proof. We need to prove only the \if" part and we may assume that X = �. Let

f�� j � < g be a decomposition of [�]m Without loss of generality we may assume

thatS�<�� 6= [�]m for every � < . Let us choose an element x0 2 [�]m and,

inductively, an element x� 2 [�]m nS�<�(�� [ fx�g) for every � with 0 < � < .

If we set �0� = fx�g[ (�� n

S�<�(�� [fx�g)) for all � < , then it is not di�cult

to check that f�0�j � < g is a partition of [�]m. By the hypothesis there exist

H 0 � � and � < such that jH 0j = � and [H 0]m � �0�. By taking H = H 0 n x� we

see that jHj = jH 0j, x� 62 [H]m and therefore [H]m � ��, as wanted. �

As we prefer decompositions to partitions, we formulate the result due to Erd}os

and Rado in the language of the monograph [17]:

Theorem 2.3. [17, Theorem 2.2.5] (2�)+ ! (�+)2� for all in�nite cardinals �.


Obviously, under GCH we have �++ ! (�+)2� and, moreover, �++ ! (�+)2� for

every � � �.

Finally, we prove an easy technical combinatorial lemma.

Lemma 2.4. If � is an in�nite cardinal of countable co�nality, then there exists an

almost disjoint family M� [�]! with degree of disjunction ! such that jMj > �.

Proof. Since � � ! = �, there exists a disjoint family D � [�]! of cardinality � such

thatSY 2D Y = �. Using Zorn's Lemma, we may extend D to a maximal (with

respect to inclusion) almost disjoint family M� [�]! with degree !. Assume that

jMj = �.

As cf(�) = !, there is an increasing chain of families Mn such that M =Sn<!Mn and jMnj < �. Since all sets from M are countable, j

SY 2Mn

Y j <

� � ! = �, hence there exists xn 2 � nSY 2Mn

Y for each n < !. Now, put

A = fxnj n < !g and �x an arbitrary Z 2M. Obviously, A is an in�nite countable

set. As there exists n0 for which Z 2 Mn0 and so xn 62 Z for each n � n0, we

obtain that jA \ Zj < !. Hence we have proved that A 62 M and M[ fAg forms

an almost disjoint family of in�nite countable sets with degree !, in contradiction

with the maximality of M. Thus jMj > �. �

3. Necessary conditions

In this section R denotes an abelian regular semiartinian ring of socle length

� + 1 with socle chain (S� j � � � + 1) throughout this section. Moreover we put

�� = gen(S�+1=S�) for each � < � (cf. Corollary 1.7) and �� is de�ned as the

(�nite) rank of the semisimple ring R=S�. As all primitive factors of R are simple,

cardinality of the set of all idempotents of R is upper bounded by 2gen(Soc(R)) by

Lemma 1.3(2) and Corollary 1.5. Clearly, the dimension sequence is determined

in such case by the cardinals ��, � � �, and by simple modules, which have the

structure of skew �elds. Applying Lemma 1.3(2) again we can describe a basic

correspondence between ranks of single slices of R.

Proposition 3.1. Suppose � � � � �.

(1) jgen(I)j � 2�0 for each ideal I of R,

(2) jh�; �ij � 2�� , in particular j�j � 2�0 ,

(3) �� 2�0 ,

(4) �� 2�� .

Proof. (1) Since every ideal of an abelian regular ring is generated by a suitable set

of central idempotents, the assertion follows immediately from Lemma 1.3(2).

(2) Suppose that I =S <� I for some strictly increasing chain of ideals I and

some in�nite ordinal �. If we �x e 2 E(I +1) n E(I ) for each < �, then j�j =

jfe j < �gj � 2�0 by Lemma 1.3(2). Now, it remains to apply this observation to

the ideal I =S�� S =S� of the factor ring R=S�.

(3) �� = gen(S�+1=S�) � gen(S�+1) � 2�0 by (1).


(4) Replacing R by R=S� the assertion follows from (3). �

Before we start to examine relationship between sets of idempotents E(S�+1=S�)

and E(S�+�+1=S�+�) for � + � � � from a combinatorial point of view, we make

an elementary observation about orthogonal idempotents.

Lemma 3.2. If � � � is an ordinal, then there exists E1 � E(S�+1) n E(S�) such

that jE1j = �� and S�+1=S� =L

e2E1e(S�+1=S�) =

Le2E1

(eR+ S�)=S�.

By applying Theorem 2.1 we improve the estimate of Proposition 3.1(3) for

� = 1.

Lemma 3.3. Suppose that GCH holds. If cf(�0) > !, then �1 � �0.

Proof. According to Lemma 3.2 we may consider a set E1 � E(S2) n E(S1) of

cardinality �1 such that S2=S1 =L

e2E1(eR + S1)=S1. Note that if f and g

are di�erent elements of E1, then fg 2 S1, hence E(fgS1) = E(fS1 \ gS1) =

E(fS1) \ E(gS1) is �nite by Lemma 1.4(3). Moreover, E(fS1) is in�nite for each

f 2 E1, thus E = fE(eS1)j e 2 E1g forms an almost disjoint family with degree of

disjunction !. By applying Corollary 1.5 and Theorem 2.1 with � = jE(S1)j and

� = !, we get �1 = jEj � � = �0. �

The previous assertion applied to R=S� yields that ��+1 � �� for an arbitrary

� < �.

If we �x a set E of non-zero orthogonal idempotents from the �-th slice of R, as

it is allowed by Lemma 3.2, for every ideal J contained in S� we may consider the

set �J = fff1; f2g 2 [E]2j f1f2 2 Jg of subsets of [E]2. By constructing a suitable

decomposition and using Theorem 2.3 we prove a limit version of Lemma 3.3.

Lemma 3.4. Suppose that GCH holds and � � � is a limit ordinal. If cf(�) < cf(�0)

and �� = �0 for every � < �, then �� 0.

Proof. Assume that gen(S�+1=S�) = �� > �0. By applying Lemma 3.2 we get

a subset E1 of E(S�+1) n E(S�) for which jE1j = �� and S�+1=S� =L

e2E1(eR +

S�)=S�. By using Corollary 1.5 we see that for every � < � there exists an increasing

chain of ideals (I�� j � < cf(�0)) of R=S� such that S�+1=S� =S�<cf(�0)

I�� and

jE(I��)j = gen(I��) < �� = �0. Obviously, for each e 2 E1 and for each � < �

there exists an ordinal �e� < cf(�0) for which eI��e� 6= 0. Let us consider a strictly

increasing co�nal trans�nite sequence f� j < cf(�)g of the ordinal �, so that

� = supf� j < cf(�)g, and put �e = supf�e� j < cf(�)g for every e 2 E1. Since

cf(�) < cf(�0), we get �e < cf(�0) by [11, Lemmas 3.6, 3.7, 3.8]. Thus eI� �e 6= 0

for every e 2 E1 and < cf(�). As jE1j > �0 = �20, there exists �0 < cf(�0)

such that jfe 2 E1 j �0 = �egj > �0. Set E2 = fe 2 E1 j �0 = �eg. Recall

that jE2j > �0 and eI� �0 6= 0 for every < cf(�) and every e 2 E2. Now, put

� = supfgen(I� �0)j < cf(�)g. Since cf(�) < cf(�0) and gen(I� �0) < �0, it follows

that � < �0.


Note that we intend to apply Theorem 2.3 to the cardinal � = max(�; cf(�)).

Since cf(�) < cf(�0) � �0, it follows that � < �0 < jE2j, hence there exists a

subset E of E2 such that jEj = �++. For every < cf(�) Let us de�ne the set

� = fff1; f2g 2 [E]2j f1f2 2 S� g for every < cf(�) and � = ; whenever

� cf(�). Obviously, for every ff1; f2g 2 [E]2 there exists < cf(�) such that

f1f2 2 S� , hence [E]2 =S <cf(�)� =

S <�� . By applying Theorem 2.3

to the decomposition f� j < �g we obtain that there are an ordinal < �

and a subset H of E of cardinality �+ such that [H]2 � � . In other words, we

have proved that f1f2 2 S� for every di�erent f1; f2 2 H. This implies thatL

f2H fI� �0 � S� +1=S� . Finally, as fI� �0 6= 0 for each f 2 H, we have

� � gen(I� �0) = jE(I� �0)j � jHj = �+, a contradiction. �

Note that the decomposition f� j < cf(�)g from the last proof is very far from

being a partition, as it is a chain. Using the argument of Lemma 2.2, we can see

that Theorem 2.3 proves about the set E much more than we actually need.

Now we can sum up the previous results.

Theorem 3.5. Suppose that GCH holds and �; � are ordinals satisfying �+ � � �.

If cf(��) > max(j�j; !), then ��+� � ��. Otherwise ��+� � �+� .

Proof. First, note that ��+� � �+� follows under GCH from Proposition 3.1. We

prove that ��+� � �� for cf(��) > max(j�j; !) by trans�nite induction on � � �.

The base step is trivial; ��+0 = ��. Suppose that ��+ � �� for all <

�. If there exists < � for which ��+ < ��, then ��+� � �+�+ � �� by

Proposition 3.1(4) and GCH. Now, suppose that ��+ = �� for all < �. If �

is non-limit, i.e. � = 0 + 1, then ��+� � ��+ 0 = �� by by Lemma 3.3 applied

to the ring R=S�+ 0 since cf(��+ 0) = cf(��) > ! by the hypothesis. Finally, if

� is a limit ordinal, the assertion follows immediately from Lemma 3.4 applied to

R=S�. �

Recall that index of a (general) ring is the supremum of the index of nilpotence

of its all nilpotent elements. Since every semiartinian ring with primitive factors

artinian is a subring of a countable product of semiartinian rings of bounded index

by [18, Lemma 1.4] which have similar properties as abelian regular semiartinian

rings (cf. [10, Chapter 7] and [18, Section 2]), we conjecture that results of the

present section remains true (and can be re�ned) for that larger class of semiartinian

rings.

4. Constructions

First, we make an easy observation on subalgebras generated by ideals.

Lemma 4.1. Let K be a �eld and I a proper ideal of a K-algebra V . If A is the

K-subalgebra of V generated by I, then I is an ideal of A and A=I �= K.

Proof. As A = K+I and I 6= A, we getK\I = 0, hence A=I �= K=(K\I) �= K. �


We intend to construct commutative semiartinian algebras over a �eld. Through-

out this section K denotes a given �eld, � a cardinal number and (R j < �) a

family of commutative, regular and semiartinian K-algebras. For each < �, we

denote with � + 1 and (S� j � � � + 1) respectively the socle length and the

socle series of R . We denote with �� the �-th projection from the direct productQ <�R onto R� .

The following technical lemmas are based on the tools developed in the paper

[6] and on Lemma 1.6.

Lemma 4.2. Let � be an in�nite cardinal. Put V =Q <�R , I =

L <�R and

� = sup <�(� + 1). Suppose that A is a K-subalgebra of V containing I.

(1) I is a semiartinian ideal of A with the socle chain (L

<� S� j � � �).

(2) If A=I is a regular ring, A is regular as well.

(3) Suppose that for every a 2 A n I and � < � there are in�nitely many < �

such that � (a) 62 S� . If A=I is regular semiartinian, then A is regular

semiartinian as well andL

<� S� is the �-th member of the socle chain

of A for every � � �.

Proof. (1) Since I is a V -module, I is an ideal of A, . Note that R as a V -module

has the same structure as an R -module, hence the lattices of A-submodules and

R -submodules of R coincide. In particular, R is a semiartinian A-module with

the socle chain (S� j � � � + 1). Thus I is a semiartinian A-module with the

socle chain (L

<� S� j � � �).

(2) It follows directly from [10, Lemma 1.3].

(3) A is commutative regular by (2) and, obviously, it is semiartinian. Applying

(1) it su�ces to show for every a 2 AnI and every � < � that a+T� 62 SocA(A=T�),

where T� is the �-th member of the socle chain of I, i.e. T� =L

<� S� .

Given a 2 A n I, let us consider the set G = f < �j � (a) 62 S� g. Choose

idempotents e 2 A and e 2 I such that aA = eA and e A = R for each 2 G

(where R is considered as a principal ideal of A). Note that ee 62 T�, since

� (a) 62 S� for every 2 G. Hence fee + T�j 2 Gg forms an in�nite subset of

E(e(A=T�)), which implies that e+T� and so a+T� is not an element of SocA(A=T�)

by Lemma 1.4(3). �

We construct a semiartinian K-algebra of socle length � + 1 for a limit ordinal

�, if there exists a suitable chain of rings of lower socle lengths.

Proposition 4.3. Let � be an in�nite cardinal, � = sup <�(� ) and �� =

gen(S�+1 =S� ) for every < � and � < � . If � > � for each < �, then there

exists a commutative regular semiartinian K-algebra A of socle length �+1 with so-

cle chain (T�j � � �+1) such that dimK(T�+1=T�) = 1 and gen(T�+1=T�) = ��

for every < � and � < � .

Proof. Applying Proposition 1.8 to the family (R j < �) we obtain a commutative

regular semiartinianK-algebra A of socle length �+1 and with socle chain (T�j � �


� + 1) such that T�+1=T� �= K and T� =L

<� S� for every � � �. Hence

gen(T�+1=T�) = gen(L

�<� S�+1�=S��) =P

�<�;�� = �� for every < �

and � < � by the hypothesis and by [11, Lemma 5.8]. �

Suppose in the rest of the section that we have constructed a commutative regular

semiartinian K-algebra R with socle chain (S�j � � �+1). A similar argument as

above allows us to construct a semiartinian K-algebra of socle length � + 2 which

has the same ranks of all but the last two slices.

Proposition 4.4. If � is an in�nite cardinal and � � gen(S�+1=S�) for every � <

�, there exists a commutative regular semiartinian K-algebra A of socle length �+2

with socle chain (T�j � � �+2) such that dimK(T�+2=T�+1) = 1, gen(T�+1=T�) =

�, and gen(T�+1=T�) = gen(S�+1=S�) for every � < �.

Proof. Put �� = gen(S�+1=S�) for each � < �. If we apply Proposition 1.8 to the

ideal R(�) of the algebra V = R�, i.e. � = � and R = R for all , we obtain a

commutative regular semiartinian K-algebra A of socle length � + 2 whose socle

chain (T�j � � � + 2) satis�es the conditions T�+2=T�+1 �= K and T� = S(�)�

for every � � � + 1. Hence gen(T�+1=T�) = 1 � � = �, and gen(T�+1=T�) =

gen((S�+1=S�)(�)) = � � �� = �� for every � < �, since � � ��. �

The �nal step of our construction uses the combinatorial argument of Lemma 2.4.

Lemma 4.5. Let � be an in�nite cardinal of countable co�nality, V = R� and

�� = gen(S�+1=S�) � � for every � < �. Then there exists a commutative regular

semiartinian K-subalgebra A of V with socle chain (T�j � � � + 2) such that

dimK(T�+2=T�+1) = 1, gen(T�+1=T�) = �+ and T� = S(�)� for each � � �.

Proof. Put I = R(�). For every C � � denote by RC the naturally de�ned principal

ideal of the ring R� and by eC the idempotent generating RC , i.e. � (eC) = 1 if

2 C and � (eC) = 0 otherwise. According to Lemma 2.4 there exists an almost

disjoint family S � [�]! with degree of disjunction ! such that jSj = �+. Now,

de�ne A as a K-subalgebra of V generated by I [ feC j C 2 Sg. Note that I is a

semiartinian ideal of A by Lemma 4.2(1) and, obviously, A = I +K +P

C2S eCK.

Since we intend to apply Lemma 4.2(3), we have to verify that A=I is regular

semiartinian and for every a 2 A n I, � � � there are in�nitely many < � such

that � (a) 62 S�.

Fix two distinct sets C;D 2 S. Note that � (i) = 0 for all but �nitely many <

� whenever i 2 I, hence eC 62 I. Moreover, since C\D is �nite eC �eD 2 RC\D � I,

which implies that (eCA+ I)=I �=A (eCK + I)=I �=A K, so (eCA+ I)=I is a simple

A-module. Thus (eC + Ij C 2 S) forms a set of non-zero orthogonal idempotents

of the factor ring A=I such that J =L

C2S eC(A=I) � Soc(A=I). Obviously,

J is in�nitely generated, hence A=I is not semisimple. Moreover, (A=I)=J is an

1-dimensional K-vector space by Lemma 4.1. Thus J = Soc(A=I) and we have

shown that A=I is semiartinian. Note that A=I is embeddable into the commutative


regular K-algebraQC2S eC(A=I)

�= K jSj and we have J �= K(jSj) by Lemma 1.3(1)

Applying Lemma 4.2(2) for an ideal K(jSj) of the algebra K jSj we obtain that A=I

is regular.

Let a 2 A n I. There exist an integer n, pairwisely distinct sets C1; : : : ; Cn 2 S,

elements k0; k1; : : : ; kn 2 K and i 2 I such that a = i + k0 � 1 +Pn

i=1 kieCi and

kj 6= 0 for at least one j. Put b = a � i. As the set f < �j � (i) 6= 0g is

�nite and b 62 I, we obtain that the set G = f < �j � (b) 6= 0; � (i) = 0g is

in�nite. Observe that � (a) = � (b) is an invertible element of R, so � (b) 62 S�

for every 2 G and � � �. Now, Lemma 4.2(3) yields that A is semiartinian

and T� = S(�)� for � � � where (T�j � � �) is the socle chain of A. Moreover,

gen(T�+1=T�) = jSj = �+, since T�+1=T� = Soc(A=I) =L

C2S eC(A=I). Finally,

T�+2=T�+1 �= (A=I)=Soc(A=I) �= K. �

Now we are able to improve the construction of Proposition 4.4, in the case

cf(�) = !.

Proposition 4.6. If � is an in�nite cardinal of countable co�nality and � �

gen(S�+1=S�) for every � < �, then there exists a commutative regular semiar-

tinian K-algebra A of socle length �+2 with socle chain (T�j � � �+2) such that

dimK(T�+2=T�+1) = 1, gen(T�+1=T�) = �+, and gen(T�+1=T�) = gen(S�+1=S�)

for every � < �.

Proof. By applying Lemma 4.5 for � = � we get a commutative regular semiartinian

K-algebra A of socle length � + 2 and with socle chain (T�j � � � + 2) such that

T�+2=T�+1 �= K, gen(T�+1=T�) = �+ and T� = S(�)� for every � < �. Finally, note

that gen(T�+1=T�) = gen((S�+1=S�)(�)) = � � gen((S�+1=S�)) = gen((S�+1=S�))

for each � < �. �

Example 4.7. By [11, Lemma 9.21] there exists an almost disjoint family on ! of

cardinality 2!. Using the argument of the proof of Lemma 4.5 for such an almost

disjoint family, we have got a commutative regular semiartinian algebra of socle

length 3 such that gen(S1) = ! and �1 = gen(S2=S1) = 2!. We have shown that

the estimation of Proposition 3.1(3) cannot be improved in general, since we have

�1 = 2�0 . Finally, let us stress that the construction need neither GCH nor the

Continuum Hypothesis.

5. Main results

The constructions of Propositions 4.3, 4.4 and 4.6 allow us to prove the main

representation theorem of this paper. Before we state it, let us remark that Proposi-

tion 4.3, which is our only tool to construct semiartinian algebras of the socle length

�+1 for a limit ordinal �, does not enable to produce an algebra with socle length

greater then the dimension of the socle whenever cf(�) = j�j, since the construction

needs the sum of at least cf(�) algebras. More generally, it is a gap between the es-

timation jh�; �ij � 2�� of Proposition 3.1(2) and the condition jh�; �ij � �� of the


following assertion, which needs the hypothesis of Proposition 4.3. It remains an

open question whether either the estimation or the construction can be improved.

Theorem 5.1. Let � be an ordinal, K a �eld and (��j � � �) a family of cardinals

satisfying for every � � � � � the conditions:

(a) �� +� if cf(��) = !, and �� otherwise,

(b) �� < ! i� � = �.

(c) jh�; �ij � ��,

Then there exists a commutative regular semiartinian K-algebra with dimension

sequence f(��; f(1;K��) j � < ��g) j � � �g where K�� = K for all � � � and

� < ��.

Proof. By Corollary 1.7 it is su�cient to show that, for every ordinal �, there exists

a commutative, regular and semiartinian K-algebra R� such that, if (S��j� � �+1)

is its socle chain, then �� = gen(S�+1;�=S��) for � < �, dimK(S�+1�=S��) = 1 and,

in addition, the conditions (a), (b) and (c) are satis�ed. We proceed by trans�nite

induction on �.

First, put R0 = K.

Given an ordinal � such that 0 < � � �, suppose that, for every � < �, an

algebra R� with the required properties exists.

Assume �rstly that � is limit and put � = cf(�). Then there exists a strictly

increasing trans�nite sequence of ordinals f� j < �g such that sup <�(� ) = �.

Note that � = cf(�) � jh�; �ij � jh�; �ij � �� for every � < � by the condition (c) of

the hypothesis, hence the existence of R� follows immediately from Proposition 4.3.

Now, let � = �+ 1 for some ordinal �. Note that �� is in�nite by (b). If ��

for every � < �, then an algebra R� exists by Proposition 4.4 where � = �� and

R = R�. Finally, suppose that there exists � < � for which �� > ��. Then �� = �+� ,

cf(��) = ! and �� = �� whenever � < � and �� > �� by (a). Moreover, �� for

each � < �, otherwise we would have �� > �� > �� , so �� > �+� , which contradicts

(a). Hence we may apply Proposition 4.6 for � = �� and R = R�. �

Note that if we have a family of �elds fK�j � � �g such that K� is a sub�eld

of K� whenever � > �, we may modify the construction of Theorem 5.1 such that

every inductively constructed ring R is K -algebra (and so K�-algebra for each

� > ), i.e. K�� in the dimension sequence of A may be replaced by K�. Finally,

note that the family fK�j � � �g can be chosen as strictly decreasing. For instance,

take K0 as the �eld of rational functions, over a �eld K, in indeterminates x� for

� < � and, if � < �, take K� as the sub�eld of K0 generated by K and the set

fx�j � < � < �g.

We are able to state the following criterion, which is a consequence of our main

results Theorem 3.5 and Theorem 5.1.

Theorem 5.2. Suppose that GCH holds. Let � be an ordinal, and (��j � � �) a

family of cardinals such that either cf(��) > jh�; �ij for each � < � or j�j � !.


Then there exists an abelian regular semiartinian ring R with dimension sequence

f(��; f(1;K��) j � < ��g) j � � �g for some �elds K�� i� the conditions (a) and

(b) of Theorem 5.1 are satis�ed for the family (��j � � �).

Proof. Let R be an abelian regular semiartinian ring with the dimension sequence

f(��; f(1;K��) j � < ��g) j � � �g. Then (b) follows immediately from Theo-

rem 1.1. Suppose that cf(��) > ! and � = �+ � � �. Then either j�j � jh�; �ij <

cf(��) or jh�; �ij � ! < cf(��) by the hypothesis. Thus we have shown that the

hypothesis cf(��) > max(j�j; !) of Theorem 3.5 is satis�ed in both cases, hence

�� . Finally, if cf(��) = ! then �� +� follows from Theorem 3.5 again,

which proves that (a) holds .

To prove the reverse implication it remains to show that the condition (c) of

Theorem 5.1 holds true, which follows from jh�; �ij < cf(��) � �� if cf(��) >

jh�; �ij and it is obvious if j�j � !. �

By applying the previous theorem and Corollaries 1.5 and 1.7 for abelian regular

semiartinian rings of countable socle lengths we obtain the �nal consequence.

Corollary 5.3. Suppose that GCH holds. The following conditions are equivalent

for a countable ordinal � and a family of cardinals (��j � � �):

(1) There exists an abelian regular semiartinian ring of socle length �+1 such

that if (S�j � � �+1) is its socle chain, then gen(S�+1=S�) = �� for each

� � �,

(2) there exists a �eld K�� for every � � �, � < �� and an abelian regular

semiartinian ring of socle length � + 1 with dimension sequence

f(��; f(1;K��) j � < ��g) j � � �g;

(3) (��j � � �) satis�es the conditions (a) and (b) of Theorem 5.1.

We conclude the paper with an example illustrating the fact that we cannot omit

GCH from the hypothesis of our results.

Example 5.4. Suppose the negation of the Continuum Hypothesis, i.e. !1 < 2!.

Denote by A1 the commutative regular semiartinian ring constructed in Exam-

ple 4.7 and by A2 a commutative regular semiartinian K-subalgebra of K!1 gener-

ated by K(!1). Since the socle length of A2 is equal to 2 and gen(Soc(A2)) = !1 by

Lemmas 4.1 and 4.2, A = A1 � A2 is a commutative regular semiartinian ring for

which gen(Soc(A)) = !1 and gen(Soc(A=Soc(A))) = 2! > !1 by Lemma 1.6. Since

cf(!1) 6= !, we have shown that the assertion of Lemma 3.3 is not true without the

hypothesis !1 = 2!.

References

[1] Baccella, G.: Semiartinian V-rings and semiartinian von Neumann regular rings. J. Algebra173 (1995), 587{612.

[2] Baccella, G.: On C-semisimple rings. A study of the socle of a ring, Comm. Algebra, 8(10)(1980), 889-909.


[3] Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans.Am. Math. Soc. 95 (1960), 466{488.

[4] Camillo V.P., Fuller, K.R.: On Loewy length of rings. Pac. J. Math. 53 (1974), 347{354 .[5] Dung N.V., Smith, P.F.: On semi-artinian V -modules. J. Pure Appl. Algebra 82 (1992),

27{37.[6] Eklof, P.C., Goodearl K.R., Trlifaj, J.: Dually slender modules and steady rings. Forum

Math. 9 (1997), 61{74.[7] Erd}os P., Rado, R.: A partition calculus in set theory. Bull. Amer. Math. Soc., 62 (1956),

427{489.[8] Fuchs, L.: Torsion preradicals and ascending Loewy series of modules. J. Reine Angew. Math.

239/240 (1969), 169{179.[9] Golan, J.S.: Torsion Theories. Longman - Harlow - Wiley, New York 1986.[10] Goodearl, K.R.: Von Neumann Regular Rings. Pitman, London 1979, Second Ed. Krieger,

Melbourne 1991.[11] Jech, T.: Set theory. The third millennium edition, revised and expanded.. Springer Mono-

graphs in Mathematics, Springer, Berlin - Heidlberg 2003.[12] Krull, W.: Zur Theorie der allgemeinen Zahlringe. Math. Ann. 99 (1928), 51{70.[13] N�ast�asescu, C., Popescu, N.: Anneaux semi-artiniens. Bull. Soc. Math. France 96 (1968),

357{368.[14] R�u�zi�cka, P., Trlifaj, J., �Zemli�cka, J.: Criteria of steadiness. In: Abelian Groups, Module

Theory, and Topology, 359{372. Marcel Dekker, New York 1998.[15] Salce, L., Zanardo, P.: Loewy length of modules over almost perfect domains. J. Algebra 280

(2004), 207{218.[16] Tarski, A.: Sur la d�ecomposition des ensembles en sous-ensembles presque disjoints. Fundam.

Math. 12 (1928), 188{205.[17] Williams, N.H.: Combinatorial Set Theory. North-Holland, Amsterdam 1977.[18] �Zemli�cka, J.: Steadiness of regular semiartinian rings with primitive factors artinian. J.

Algebra, 304 (2006), 500{509.


Katedra algebry MFF UK, Sokolovsk�a 83, 186 75 Praha 8, Czech Republic

G. COMMUTATIVE TALL RINGS

TOM�A�S PENK AND JAN �ZEMLI�CKA

Abstract. A ring is right tall if every non-noetherian right module containsa proper non-noetherian submodule. We prove a ring-theoretical criterionof tall commutative rings. Besides other examples which illustrate limits ofproven necessary and su�cient conditions, we construct an example of a tallcommutative ring that is not max.

Many conditions of the category of all modules over a ring can be easily expressed

in the language of the ring theory. A typical example of such correspondence be-

tween classes of modules and the ring structure is presented by perfect rings, which

are characterized in Bass' paper [2] by both ring-theoretic and module-theoretic

conditions. In particular, recall that every nonzero right module over a right per-

fect ring contains a maximal submodule. Rings over which every nonzero right

module contains a maximal submodule are said to be right max (or almost noether-

ian) and they are widely studied by many authors from various points of view and

with di�erent motivations [3, 4, 5, 6, 7, 8, 11, 15]. Nevertheless, a ring-theoretic

criterion of max rings is available only for a few interesting classes of rings (e.g. for

p.i.-rings in [11]).

The notion of a tall module goes back to paper [13] and it is de�ned as a mo-

dule M which contains some submodule N such that both M=N and N are non-

noetherian. A ring is called right tall if every non-noetherian right module is tall.

Although [13, Theorem 2.7] presents a nice characterization of right tall rings using

the notion of Krull dimension of all modules, a general ring-theoretic necessary and

su�cient condition hasn't been known yet. It is not hard to see that every right

max ring is right tall, and John Clark in [5] asked if there was a di�erence between

the classes of all tall and all max rings.

In this paper we present several necessary conditions as well as several su�cient

conditions for commutative tall rings, which are easily applicable to many natural

classes of rings. Namely, ifTi Ji is not a prime ideal for every countable decreasing

chain of ideals fJig of a commutative ring R such that J1 is maximal, J1Ji � Ji+1,

and R=Ji is artinian for each i, we prove in Theorem 2.6 that R is tall. On the other

hand, if R is tall, then for every non-idempotent maximal ideal I such that R=Ii is

artinian for each i, the intersectionTj I

j is not a prime ideal (Proposition 2.9). As

a consequence we prove a ring-theoretic criteria of tallness for general commutative

(Theorem 2.12) and for noetherian commutative rings (Proposition 2.10). The last

1991 Mathematics Subject Classi�cation. 16P70 (13E10).Key words and phrases. tall ring, max ring, non-noetherian artinian module.

109

110 TOM�A�S PENK AND JAN �ZEMLI�CKA

section contains examples illustrating the limits of both necessary and su�cient

conditions (Examples 3.4, 3.7) and an example of a tall non-max commutative ring

(Example 3.2).

1. Preliminaries

Throughout the paper, a ring R means an an associative ring with unit and a

module means a right R-module. For a module M denote by E(M) an injective

envelope, J(M) the Jacobson radical, Ann(M) the annihilator and Soc(M) the

socle of M .

Given a module M , the Loewy (or socle) chain of M means an increasing chain

of submodules (S�(M) j � � 0) de�ned by the following rules: set S0(M) = 0 and,

recursively, S�+1(M)=S�(M) = Soc(M=S�(M)) for each ordinal � and S�(M) =

=S�<� S�(M) if � is a limit ordinal. The �rst ordinal � such that S�(M) =

S�+1(M) is called the socle length of M and one says that M is semiartinian if

S�(M) =M . Note that a submodule and a factor-module of a semiartinian module

M are semiartinian and S�(N) = S�(M)\N for each � and each submodule N of

M . A ring is called right semiartinian if it is semiartinian as a right module. For

other basic properties of semiartinian modules and rings we refer to [10]. Recall

that the length of a module M is the length of a composition series of M , i.e. n

such that 0 = M0 �M1 � � � � �Mn = M where Mi+1=Mi is simple. Obviously, if

M is of length n, then it is semiartinian of socle length � n.

We recall the general criterion of tall rings proven by Sarath:

Theorem 1.1. [13, Theorem 2.7] The following conditions are equivalent for a ring

R:

(1) R is right tall,

(2) every non-noetherian module has a proper non-noetherian submodule,

(3) every module with Krull dimension is noetherian.

As a consequence we obtain (cf. [5, p. 31]):

Corollary 1.2. Every right max ring is right tall.

Note that closure properties of the class of all tall rings are similar to those of

the class of all max rings:

Lemma 1.3. The class of all right tall rings is closed under factors, �nite products,

and Morita equivalence.

Proof. Since a non-tall module over a factor ring R=I has a natural structure of a

non-tall module over the ring R, we see that a factor of every tall ring is tall.

Let R =Qn

i=1Ri be a product of rings which is not right tall. Denote by ei 2 R

the central idempotent satisfying (ei)j = �ij and �x a non-noetherian moduleM for

which every proper submodule is noetherian. Such a module exists by Theorem 1.1.

G. COMMUTATIVE TALL RINGS 111

Then M =Ln

i=1Mei and there exists j for which Mej is not noetherian, hence

Mej =M and Rj is not right tall again by Theorem 1.1.

As the existence of a non-tall module implies the existence of a non-tall module

over a Morita equivalent ring, the last closure property is clear. �

Moreover, if R is a tall subring of a commutative ring S and S is �nitely generated

as an R-module, then S is tall by [13, Theorem 2.11].

Note that every right non-tall ring is not max, and over a right non-max ring

there exists a non-zero module which contains no maximal submodule. Now we

recall several easily veri�able properties of such modules, which we shall apply

later.

Lemma 1.4. Let M be a module over a commutative ring R, N a proper submodule

of M and r 2 R nAnn(M). If J(M) =M , then

(1) J(M=N) =M=N is not noetherian,

(2) J(rM) = rM is not noetherian,

(3) if M is not tall then M=N and Mr are not tall modules

2. Ring structure

We begin this section by an easy observation that for each prime number p the

Pr�u�er p-group Zp1 forms a natural example of a non-tall Z-module. The following

example due to Sarath [13, Proposition 2.8] shows that a similar construction of

non-tall modules works also over an arbitrary polynomial commutative ring:

Example 2.1. Let R be a commutative ring and I a maximal ideal of R. Then

F = R=I is a �eld and F [x] has a natural structure of a cyclic module over the

polynomial ring R[x]. For every n put Cn = F [x]=xnF [x] and note that xCn+1 �=

Cn, hence we have a natural embedding in : Cn ! Cn+1. It is easy to see that

the inverse limit C of C1i1,! C2

i2,! : : : is a uniserial artinian module which is not

noetherian, hence C presents an example of a non-noetherian non-tall R[x]-module.

We make an easy technical observation about semiartinian modules including

the previous examples.

Lemma 2.2. Let M be a semiartinian module of an in�nite socle length such that

every proper submodule ofM is of �nite length. ThenM is artinian, non-noetherian

and non-tall.

Proof. M is not noetherian because fSj(M)j j < !g forms a strictly increasing

chain of submodules. If M0 ) M1 ) : : : is a strictly decreasing chain of submod-

ules, then M1 is artinian by the hypothesis, and hence the sequence terminates

after �nitely many steps. Finally, N is non-tall since every proper submodule is

noetherian. �

The previous observation about in�nitely generated semiartinian modules allows

us to generalize the construction of the uniserial examples of non-tall modules:


Lemma 2.3. If M is a non-noetherian semiartinian module for which every Loewy

chain member is artinian, then there exists a non-noetherian artinian submodule of

M which is not tall.

Proof. Note that every submodule of a semiartinian module is semiartinian and

that S!(M) is a non-noetherian submodule of M . Put

N = fN � S!(M)j Sj+1(N)=Sj(N) 6= 0 8j < !g:

Obviously, S!(M) 2 N , hence N 6= ;. Suppose (N� , � < �) is a decreasing chain of

modules from N . Let j < !. Since Sj(N�) = Sj(M)\N� is an artinian module for

each � < � by the hypothesis, there exists �j < � such that Sk(N�) = Sk(N�j ) for

every k � j + 1 and � > �j , hence Sj+1(N�)=Sj(N�) = Sj+1(N�j )=Sj(N�j ) 6= 0 for

each � > �j . It implies that Sj+1(N�j ) �T� N� and Sj+1(

T� N�)=Sj(

T� N�) 6= 0

for every j and soT� N� 2 N .

Using a Zorn's lemma argument we obtain a minimal submodule N with respect

to inclusion such that Sj+1(N)=Sj(N) are nonzero for all j. Clearly, N has an

in�nite socle length. If P is a proper submodule of N , then there exists j such that

Sj+1(P )=Sj(P ) = 0 by minimality of N , thus P � Sj(M) is artinian. By applying

Lemma 2.2 we get that N is artinian, non-noetherian and non-tall. �

As every artinian module is semiartinian and submodules of artinian modules

are artinian as well, we obtain

Corollary 2.4. Every non-noetherian artinian module contains a non-tall submod-

ule.

The following technical lemma shows that over a non-tall commutative ring there

exists a module possessing a similar structure of the lattice of submodules as Zp1 .

Lemma 2.5. If R is a commutative non-tall ring, then there exists a non-tall

module M such that

(1) Ann(M) is a prime ideal,

(2) S = Soc(M) is a simple essential submodule of M and for every j < !

there exists a natural number nj such that Sj+1(M)=Sj(M) �= Snj ,

(3) M is semiartinian of socle length ! and every proper submodule of M has

�nite length,

(4) M is non-noetherian and artinian,

(5) there are elements x 2 R and sj 2M , j < !, such that sj+1x = sj for each

j and M =Sj sjR,

(6) there are elements mj 2 M , j < !, such that mjR � mj+1R for each j,

M =SjmjR and Sj(mjR) = Sj(M).

Proof. By Theorem 1.1 there exists a non-noetherian module N such that every

proper submodule of N is noetherian. Since every nonzero factor of N contains

no maximal submodule by Lemma 1.4, it is a non-tall in�nitely generated module.


Pick 0 6= u 2 N and let I be a maximal ideal containing Ann(u). Then the

natural epimorphism uR ! R=I can be extended to a nonzero homomorphism

� : N ! E(R=I). Observe that M = �(N) is a non-tall in�nitely generated module

and Soc(M) = fmj mI = 0g �= R=I. Moreover, note that aR �= R=Ann(a) is a

noetherian module, hence R=Ann(a) is a noetherian ring for every a 2M .

(1) Let rs 2 Ann(M). Assuming that r =2 Ann(M) then Mr � M is non-

noetherian by Lemma 1.4, so Mr =M and 0 =M(rs) = (Mr)s =Ms.

(2) and (3) Put Mj = fm 2 M j mIj = 0g. Obviously, S0(M) = M0 ( M1 =

S1(M) and Mj � Sj(M). We will prove by induction on j > 0 that Sj�1(M) (

Sj(M) (M and that Mj = Sj(M) is a module of �nite length:

As Sj(M) is of �nite length by the induction hypothesis and since M is not noe-

therian, Sj(M) 6=M . Moreover, Sj+1(M)=Sj(M) is a �nitely generated semisimple

module, hence Sj+1(M) is of �nite length as well.

We have noted that for every a 2M nMj the ring R=Ann a is noetherian, which

implies that there exists a maximal ideal J with respect to inclusion satisfying

the condition 9b 2 aR nMj : bJ � Mj . Fix such b and J . Let i 2 I. We will

show that i 2 J . Assume that i =2 J which implies that bi 2 M n Mj . By

the induction hypothesis Mj=Mj�1 is essential in Mj+1=Mj�1, hence there exists

r 2 R for which bir 2Mj nMj�1. Note that br =2Mj since bri =2Mj�1. Moreover,

br(J + iR) = bJr + briR � Mj . By the maximal property of J we see that

J + iR = J , a contradiction. Hence i 2 J . This proves I � J where I is maximal

and J is a proper ideal of R, so I = J . Since b 2 Sj+1(M) nMj , we obtain that

Sj(M) ( Sj+1(M).

Now, if m = m+Mj 2 Soc(M=Mj) = Sj+1(M)=Mj , then Ann(m) is a maximal

ideal and mAnn(m) �Mj , thus Ann(m) = I and m 2Mj+1.

We have proven that Sj(M) ( Mj+1 = Sj+1(M) 6= M and Mj+1 has �nite

length for each j. Since M0 � M1 � : : : forms a strictly increasing chain of

submodules andM contains no proper non-noetherian submodule, we getSjMj =

M . Finally, if P is a proper submodule of M , then P is noetherian, so there exists

j such that P �Mj , which proves that P is of �nite length.

(4) By (3) we may apply Lemma 2.2.

(5) Let s0 be a generator of Soc(M) and x 2 I nAnn(M). Since Mx =M by (3)

and Lemma 1.4, we can construct a sequence sn 2 M such that sk = sk+1x. It is

easy to see that skR � sk+1I 6= sk+1R, henceSk skR is a non-noetherian module

and soSk skR =M .

(6) As Mk is noetherian, there exist indices kj > kj�1 for which Mj � skjR.

Now it remains to put mj = skj . �

By applying Lemma 2.5 we are ready to formulate su�cient conditions for the

ring structure of commutative tall rings.

Theorem 2.6. Let R be a commutative non-tall ring. Then there exists a maximal

ideal I and a sequence of ideals I = J1 � J2 � : : : such that


(1) IJi � Ji+1 for each i,

(2) R=Ji is artinian for each i,

(3)Ti Ji is a prime ideal,

(4) R=Tn I

n is not a tall ring.

Proof. Set Ji = Ann(Si(M)) where M is from Lemma 2.5.

(1) IJi � Ji+1, because Si+1(M)I � Si(M) by Lemma 2.5(2).

(2) For g1; : : : ; gk generators of Si(M) de�ne the homomorphism f : R!Q

j gjR

by f(r) = (g1r; : : : ; gkr). Then Ker(f) = Ji and R=Ker(f) �= Im(f) is artinian sinceQ

j gjR is artinian.

(3) From M =Si Si(M) we have Mr = 0 i� Si(M)r = 0 for all i. Hence

Ann(M) =TiAnn(Si(M)) =

Ti Ji is a prime ideal.

(4) SinceTn I

n �Ti Ji, M has the structure of an R=

Tn I

n-module. �

We can formulate the following consequence of Theorem 2.6(4) and Lemma 1.3:

Corollary 2.7. A commutative ring R is tall i� R=Tn I

n is tall for every maximal

ideal I.

Now, we prove several necessary conditions for commutative tall rings.

Lemma 2.8. If R is a commutative ring and I is a maximal ideal such that Ij�1=Ij

is nonzero artinian for each j, then E(R=I) contains a non-noetherian semiartinian

module M for which every Loewy chain member is artinian.

Proof. Set E1 = R=I = E(R=IR=I), Ej = E((Ej�1)R=Ij ) = E(R=IR=Ij ) and M =Sj Ej . Notice that R=Ij is local with I=Ij the unique maximal ideal and that

the socle length of R=Ij is equal to j by the hypothesis. Hence Sj(M) = fm 2

M j mIj = 0g and Ej ( Ej+1. Evidently EjIj = 0, Ej � Sj(M) and S1(M) =

R=I. Because E(R=IR) is injective we have Ej � E(R=IR) and, consequently,

M � E(R=IR). From S1(M) E Sj(M), S1(M) � Ej and R=Ij-injectivity of Ej we

obtain an embedding Sj(M) ,! Ej . The length of Ej is at least j (E0 ( E1 : : : (

Ej�1 ( Ej) and �nite by [9, Corollary(3.85)] therefore Ej = Sj(M) is artinian for

all 0 < j < ! and M is not noetherian. �

Proposition 2.9. Let R be a commutative ring and I a maximal ideal. If R=Ij is

artinian for each j andTj I

j is a non-maximal prime ideal then R is not tall.

Proof. SinceTj I

j is prime but not maximal, R=Tj I

j is an integral domain but

not a �eld and it follows easily from this that Ij ) Ij+1 for each j. As there exists

by Lemma 2.8 a non-noetherian semiartinian module such that every member of

its Loewy chain is artinian we obtain that R is non-tall by Lemma 2.3. �

As an easy consequence of the last proposition we get the well-known fact that

there exist non-noetherian artinian modules over every noetherian commutative

domain which is not a �eld. Combining this fact with the Krull intersection theorem

gives us a criterion of tallness for the class of all commutative noetherian rings.


Proposition 2.10. Let R be a commutative noetherian ring. Then the following

conditions are equivalent:

(1) R is tall,

(2) R contains no non-maximal prime ideal,

(3) R is artinian.

Proof. (1)!(2) If R contains a non-maximal prime ideal P and I � P is maximal,

then according to the Krull intersection theorem [12, Theorem 8.10(ii)] we haveTj(I=P )

j = P and R=Ij is artinian for all j by Hopkins' theorem. Then R=P is

not tall by Proposition 2.9.

(2)!(3) Denote by N(R) the nilradical of R, i.e. the intersection of all prime

ideals and note that N(R) is nilpotent since R is commutative noetherian and that

N(R) = J(R) by the hypothesis. Moreover, there exist maximal ideals I1; : : : ; In

such that J(R) = N(R) =TIj by [12, Theorem 6.5], hence R=J(R) is semisimple.

The conclusion follows by Hopkins' theorem.

(3)!(1) This follows from Corollary 1.2 since every artinian ring is max. �

A ring is called valuation provided it is commutative and its lattice of ideals is

linearly ordered.

Since R=Tn I

n is necessarily noetherian for every valuation ring R and its max-

imal ideal I, we get an easy consequence of Corollary 2.7 and Proposition 2.10:

Corollary 2.11. A valuation ring is tall i� it is artinian or it has in�nitely gene-

rated Jacobson radical.

The following criterion shows that the Pr�u�er p-group is a typical example of a

non-tall module.

Theorem 2.12. The following conditions are equivalent for a commutative ring

R:

(1) R is not tall,

(2) there exists a non-noetherian artinian module,

(3) there exists an artinian module M , elements x 2 R and mj 2M such that

mj+1x = mj and mj+1 =2 mjR for each j and M =SjmjR.

(4) there exists a semiartinian module M and a sequence of elements mj 2M ,

j < !, such that Sj(mjR) = Sj(M) is �nitely generated, mjR ( mj+1R

for each j, and M =SjmjR,

(5) there exists a sequence of ideals Jj of R and elements xj 2 R such that R=Jj

is artinian, Jj+1 ( Jj, xjr 2 Jj+1 i� r 2 Jj and the length of Sj(R=Jj) is

equal to the length of Sj(R=Jk) for each j � k < !.

Proof. (1)!(3) follows from Lemma 2.5(4),(5).

(3)!(2) Since every submodule of M is artinian, there exists a non-noetherian

artinian module by Lemma 2.3.

(2)!(1) This is an immediate consequence of Corollary 2.4


(1)!(4) is proved in Lemma 2.5(3),(6).

(4)!(2) This is shown in Lemma 2.3.

(4)!(5) Put Jj = Ann(mj) and �x xj such that mj+1xj = mj . Then R=Jj �=

mjR is artinian and Jj+1 = Ann(mj+1) � Ann(mj+1xj) = Ann(mj) = Jj . More-

over, xjr 2 Jj+1 = Ann(mj+1) i� rmj = xjrmj+1 = 0 i� r 2 Ann(mj) = Jj . Note

that Sj(mjR) = Sj(M) and Sj(mjR) � Sj(mkR) � Sj(M) for every k � j, hence

Sj(R=Jj) �= Sj(R=Jk).

(5)!(4) Since the map ij : R=Jj ! R=Jj+1 de�ned by the rule ij(r + Jj) =

xr + Jj+1 is a monomorphism, the inverse limit of R=J1i1,! R=J2

i2,! : : : forms a

module M where mj 2M are the images of the elements 1 + Jj 2 R=Jj . �

3. Examples

Recall that a ring R is (Von Neumann) regular if for each x 2 R there is y 2 R

for which xyx = x. It is easy to see that every prime ideal of a regular ring is

necessarily maximal. Now, we generalize [13, Example (1), p.335]:

Lemma 3.1. Let R be a commutative ring and J an ideal. If R=J is regular and

every element of J is nilpotent, then R is tall.

Proof. Since R contains no non-maximal prime ideal, the conclusion follows imme-

diately from Theorem 2.6. �

Recall that by Corollary 1.2 every max ring is tall. We show that the converse

is not true, which answers negatively the question of John Clark [5, Section 4].

Example 3.2. Let F be a �eld and X = fx1; x2; : : : g an in�nite countable set of

variables. Put I =P

i x2iF [X] and R = F [X]=I. Denote Xi = xi+ I and de�ne an

ideal J =P

iXiR. Then J is an ideal in which every element is nilpotent, since

X2i = 0 and R is commutative. As R=J �= F , the hypothesis of Lemma 3.1 is

satis�ed, hence R is tall.

Moreover, J is a nil maximal ideal of R, thus it is the Jacobson radical of R.

Since X1 � : : : �Xn 6= 0 for every n, J is not T-nilpotent, hence R is not a max ring

by [1, Remark 28.5].

Since the ring of p-adic integers is a noetherian valuation domain, it is not tall

by Corollary 2.11. By using a similar argument, we show that in�nite products of

tall rings need not be tall.

First, recall that a system F of subsets of a set X is called an ultra�lter if ; =2 F ,

it is closed under �nite intersections and all oversets, and for each G � X either

G 2 F or X n G 2 F . We make some observations about ring products using an

idea of [14, Lemma 2.2]:

Lemma 3.3. Let R be a commutative ring, F an ultra�lter on !, and mR a

principal maximal ideal of R such that Rn = R=mnR is a valuation ring. Put

M = ff 2Q

n<! Rnj (9F 2 F)(8n 2 F ) : fn 2 mRng. Then M is a maximal ideal


ofQ

nRn, the ring (Q

nRn)=Mk is valuation for each k and

TkM

k is a prime

ideal.

Proof. First, we show that Mk�1=Mk is a simple module for an arbitrary k. For

simplicity of the notation we will denote by m the elements m +mnR 2 R=mnR

for each n, as well as the element m � 1 2Q

nRn. Moreover, put M0 = R.

Let r 2 Mk�1 nMk. Then G = fn < !j rn 2 mkRng =2 F and there exists

F 2 F such that rn 2 mk�1Rn for each n 2 F . Hence H = F \ (! n G) 2 F and

!nH =2 F since F is an ultra�lter. Note that there exists an invertible tn 2 Rn such

that rn = mk�1tn for each n 2 H, and de�ne a; b 2Q

nRn by the rules an = 0

and bn = t�1n if n 2 H and an = mk�1 and bn = 0 whenever n =2 H. Clearly,

a + rb = mk�1 and a 2 Mk, so we have proved that Mk + rQ

nRn = Mk�1.

Hence Mk�1=Mk is a simple module and M is a maximal ideal. It implies that

(Q

nRn)=Mk is a valuation ring. Finally, if a; b =2

TkM

k, then there is a k for

which a; b =2Mk , hence ab =2M2k because (Q

nRn)=M2k is valuation. �

Example 3.4. Let p be a prime number, Zpn denote the cyclic group of pn elements

and S =Q

n<! Zpn . Then by Lemma 3.3 the ring S contains a maximal ideal I

such that S=In is valuation, hence artinian for every n. By applying Proposition 2.9

we obtain that S is not tall, however every ring Zpn is tall.

Nevertheless, the product of artinian rings can give us new examples of tall

modules.

Proposition 3.5. Let � be a cardinal and R� commutative artinian rings, � < �.

If there exists n < ! such that J(R�)n = 0 for each � < �, then

Q�<�R� is tall.

Proof. Since (Q

� J(R�))n = 0 and

Q�R�=

Q� J(R�) �=

Q�R�=J(R�) is a regular

ring,Q

�<�R� is tall by Lemma 3.1. �

Note thatQ

�<�R� from the previous proposition is even max by [5, Theo-

rem 3.4].

However, it is not hard to describe which localization of non-tall rings are not

tall. We present only a few examples:

Example 3.6. Localization of the ring Z in the maximal ideal pZ for each prime

p and the localization of the polynomial ring F [x] over any �eld F in the maximal

ideal xF [x] are noetherian domains, so they are not tall by Proposition 2.10.

Finally, we illustrate that ideals Jn from Theorem 2.6 cannot be replaced by Jn1in general.

Example 3.7. Consider the maximal ideal M =P

n xnF [X] of the polynomial

ring F [X] in countably many variables X = fx1; x2; : : : g over a �eld F and let R

denote the localization of F [X] in M . Put PA =P

j2A xjR and note that PA is a

prime ideal for each A � ! and P! is the unique maximal ideal of R. Obviously, R

is not noetherian, and R=P!nf0g �= F [x0](x0F [x0]). As the localization F [x0](x0F [x0])


is non-tall by Example 3.6 and every factor of a tall ring is tall by Lemma 1.3, R

is not tall. However, it is easy to �nd ideals Ji =Pn

j=0 xijR +

Pj>i xjR ensured

by Theorem 2.6. All factors M j=M j+1 are in�nitely generated modules, hence we

cannot use Proposition 2.9 to construct a non-tall module.

References

[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, 2nd edition, New York1992, Springer.

[2] H. Bass, Finitistic dimension and a homological generalization of semiprimary rings. Trans.Am. Math. Soc. 95 (1960), 466{488.

[3] V. P. Camillo, On some rings whose modules have maximal submodules, Proc. Amer. Math.Soc. 50 (1975), 97{100.

[4] S. Charalambides, Stelios; J. Clark, Max modules relative to a torsion theory, J. AlgebraAppl. 7 (2008), No. 1, 21-45.

[5] J. Clark, On max modules, Proceedings of the 32nd Symposium on Ring Theory and Repre-sentation Theory, Tokyo 2000, 23{32.

[6] C. Faith, Rings whose modules have maximal submodules, Publ. Mat. 39 (1995), 201{214.[7] Y. Hirano, On rings over which each module has a maximal submodule, Comm.Algebra 26

(1998), 3435{3445.[8] L. A. Koifman, Rings over which every module has a maximal submodule, Mat. Zametki 7

(1970), 359{367; (transl.) Math. Notes 7 (1970), 215{219.[9] T.Y. Lam Lectures on Modules and Rings, Springer-Verlag, New York, 1999.[10] C. N�ast�asescu, N. Popescu, Anneaux semi-artiniens. Bull. Soc. Math. France 96 (1968),

357{368.[11] V.T.Markov, B-rings with a polynomial identity, J. Sov. Math. 31 (1985), 3238{3243.[12] H. Matsumura, Commuative Ring Theory, Cambridge 1989, Cambridge University Press.[13] B. Sarath, Krull dimension and noetherianness, Illinois J. Math. 20 (1976), 329{335.[14] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents. Abelian groups and

modules (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995, 467{473.[15] A. A. Tuganbaev, Rings whose nonzero modules have maximal submodules, J. Math. Sci.

109, 1589-1640.





H. MOD-RETRACTABLE RINGS

M. TAMER KOS�AN AND JAN �ZEMLI�CKA

Abstract. A right module M over a ring R is said to be retractable ifHomR(M;N) 6= 0 for each nonzero submodule N of M and the ring R is(�nitely) mod-retractable if every (�nitely generated) right R-module is re-tractable. We proved for every �nite group G that M R RG is a retractableRG-module i�MR is retractable. Some comparisons between max rings, semi-artinian rings, perfect rings, noetherian rings, nonsingular rings and mod-retractable rings are investigated. In particular, we prove ring-theoretical cri-teria of right mod-retractability for classes of all commutative, left perfect andright noetherian rings.

1. Introduction

A module M is called retractable if there exists a nonzero homomorphism into

every nonzero submodule N � M , i.e., HomR(M;N) 6= 0 for every nonzero sub-

module N of M . The notion of a retractable module was introduced by Khuri in

[11] and it have been studied extensively by many authors (see for example, [8],

[12], [13], [14],[19],[20], [21], [22], [23], [27]). Recently, Ecevit and Ko�san [4] and in-

dependently Haghany, Karamzadeh, and Vedadi [9] introduced the concept of right

(�nitely) mod-retractable rings de�ned as rings over which every (�nitely generated)

right module is retractable.

The natural notion of the group module over a group ring was introduced and

studied by Ko�san-Lee-Zhou in [10]. It is easy to show (Lemma 2.1) that the class

of all group modules by group G over a ring R coincides with the image of the

tensor functor � R RG. Note that this subcategory plays an important role in

the category of all RG-modules. Section 2 of this note deals with the transfer of

properties of retractable modules between a right R-module and its group RG-

module. It is shown that MG is a retractable RG-module if and only if M is a

retractable R-module.

In Section 3, we investigate some comparisons between max rings, semiartinian

rings, perfect rings, noetherian rings, nonsingular rings and mod-retractable rings.

We characterize mod-retractable rings as rings whose all torsion theories are hered-

itary. As a consequence, we prove that a commutative ring is mod-retractable if

and only if it is semiartinian. Moreover, we show that a left perfect ring is mod-

retractable if and only if it is isomorphic to the ringQ

i�kMni(Ri) for a �nite

system of both left and right perfect local rings Ri, i � k. This result illustrates

2000 Mathematics Subject Classi�cation. 16D50.The second author is supported by research project MSM 0021620839, �nanced by M�SMT.

119

120 M. TAMER KOS�AN AND JAN �ZEMLI�CKA

limits of the construction of new examples of mod-retractable rings proved in [4,

Theorem 8] as �nite products of matrix rings over mod-retractable rings. Namely,

applying this procedure on the class of two-sided perfect local rings we get all exam-

ples of left perfect right mod-retractable rings. In case R is a right noetherian ring,

then it is shown that R is right mod-retractable if and only if R �=Q

i�kMni(Ri)

for a system of a local right artinian rings Ri, i � k.

Throughout this paper, we assume that R is an associative ring with unity, M

is a unital right R-module and G is a group. In the following the symbols, \�"

will denote a submodule, \�d" a module direct summand and \E" an essential

submodule. The notations J(M) means Jacobson radical of a module M , N(R) is

the prime radical of R and E(M) means an injective envelope of M . The group

ring of G over R is denoted by RG. We will refer to [1] and [24] for all unde�ned

notations used in the text.

2. Retractability for group modules

In this section we start to investigate retractable modules over group rings. As

the �rst step, which allows us to produce new examples of retractable modules, we

de�ne the notion of a group module.

Hereafter G is a group and M is a module over a ring R.

Let MG denote the set all formal linear combinations of the formP

g2Gmgg

where mg 2M and mg = 0 for almost all g.

For elementsP

g2Gmgg;P

g2G ngg 2MG andP

g2G rgg 2 RG;Xg2G

mgg =Xg2G

ngg i� mg = ng for all g 2 G

Xg2G

mgg +Xg2G

ngg =Xg2G

(mg + ng)g

(Xg2G

mgg)(Xg2G

ngg) =Xg2G

(kg)g where kg =Xhh0=g

mhr0h:

Under the operations de�ned above,MG has a structure of a right RG-module and

it is said to be the group module over the group G (see [10]). If we identify every

element m 2M with m � 1 2MG, we obtain M as an R-submodule of MG, where

1 hereafter denotes the identity element of G. Recall that 1 of G may be identi�ed

with the identity element of RG in natural way.

The following claim shows that the class of all group RG-modules for a ring R

and a group G forms precisely the image of the tensor functor �R RG. We will

use this natural description of category of group modules to construct new large

classes of retractable modules.

Lemma 2.1. If MG is the group module, then MG �=RG M R RG.

H. MOD-RETRACTABLE RINGS 121

Proof. Clearly, there exists an RG-homomorphism ' :M RRG!MG satisfying

the rule '(P

imi P

g rgig) =P

g;i(mrgi)g =P

g(P

imrgi)g. Now, it is easy to

see that ' is onto MG. Finally, ' is injective since RG is a free left R-module. �

The map MG ! M ,Pmgg !

Pmg, is an R-homomorphism and is denoted

by "M . The kernel of "M is denoted by 4(M). Thus, "R : RG ! R is the usual

augmentation map.

Lemma 2.2. Let MG be the group module of G by M over RG. Then for any

x 2 MG and any � 2 RG, "M (x�) = "M (x)"(�). In particular, "M is an R-

homomorphism and "R is a ring homomorphism.

Proof. Write x =P

g2Gmgg and � =P

g2G rgg. Then,

"M (x�) =P

g2G

�Phh

0=gmhrh0

�=�P

mg

��Prg

�= "M (x)"(�):

�

As "R is onto R and classes of (�nite) mod-retractable rings are closed under

taking factors, we get the following observations:

Corollary 2.3. If RG is a (�nitely) mod-retractable ring then R is (�nitely) mod-

retractable.

Proposition 2.4. Let R � S be an extension of rings and S a �nitely gener-

ated free left R-module. Suppose that NR is a submodule of a module MR. Then

HomR(M;N) 6= 0 i� HomS(M R S;N R S) 6= 0.

Proof. First we note that

HomS(M R S;N R S) �= HomR(M;HomS(S;N R S))�= HomR(M;N R S)

by [24, Proposition I.9.2]. Since RS is a free module, there exists a natural number

n such that RS �= Rn. Now

HomR(M;N R S) �= HomR(M;N R Rn)

�= HomR(M;N R R)n

�= HomR(M;N)n;

which implies that HomS(MRS;NRS) 6= 0 if and only if HomR(M;N) 6= 0. �

Recall that MG �=RG M R RG by Lemma 2.1. As the group ring RG is

free as both left and right R-modules, we get the following easy consequence of

Proposition 2.4.


Corollary 2.5. Let N be a submodule of a right R-moduleM . Then HomR(M;N) 6=

0 if and only if HomRG(MG;NG) 6= 0.

Lemma 2.6. Let R � S be an extension of rings, S a �nitely generated free left R-

module, M an S-module, and N an R-submodule of M R S. Then every nonzero

� 2 HomR(M;N) can be extended to a nonzero S-homomorphism of M R S into

NS.

Proof. Let � 2 HomR(M;N) and � 6= 0. Obviously, � S is an S-homomorphism

of M R S into N R S. Moreover, a mapping : N R S ! NS de�ned by the

rule (P

i ni si) =P

i nisi is an S-homomorphism as well, hence � = � S is

the required homomorphism of M R S into NS. Finally, we can easily see that

�(m 1) = (�(m) 1) = �(m) for each m 2M , i.e. 6= 0. �

Theorem 2.7. Suppose that R is a subring of a ring S, M an R-module and

fe1; : : : ; eng a free base of S as a right R-module such that eir = rei for all i and

r 2 R. Then M R S is a retractable S-module if and only if M is retractable.

Proof. Note that fe1; : : : ; eng is also a free base of S as a left R-module. As RS

is a projective module, the functor � R S is exact, hence the direct implication

follows immediately from Proposition 2.4.

Suppose that M is a retractable R-module and �x a nonzero S-submodule P of

M R S. We have to show that HomRG(M;P ) 6= 0. For each � 2M R S de�ne

�(�) = fF � f1; : : : ; ng j 9m1; : : :mn 2M : � =Xi2F

mi eig:

It is easy to see that �(�) 6= ;, hence we may put s(�) = minfjF j j F 2 �(�)g.

Now, we can choose a nonzero element � 2 P with a minimal (nonzero) value of

s(�). Thus there exist di�erent numbers i1; : : : ; is(�) � n and nonzero elements

m1; : : : ;ms(�) 2 M such that � =Pk

j=1mj eij . If there are r 2 R and j � s(�)

for which mj eijr = mjr eij = 0, then s(�r) < s(�), hence �r = 0 due to the

minimality of s(�). Thus the annihilators of all m1; : : : ;ms(�) coincide and cyclic

R-modules �R and miR are R-isomorphic. As M is retractable, there exists a

nonzero R-homomorphism of M into �R. Now, applying Lemma 2.6 for N = �R,

we obtain a nonzero S-homomorphism of M R S to �S � P , which �nishes the

proof. �

Recall that the group ring RG is right perfect i� R is a right perfect ring and G

is a �nite group by [26, Theorem]. As an immediate consequence of Theorem 2.7

we have the following similar result:

Corollary 2.8. Let M be an R-module and G a �nite group. Then MG is a

retractable RG-module i� MR is retractable.


However every tensor product M S of a retractable R-module M and a ring

extension satisfying the hypothesis of Theorem 2.7 has to be retractable over S,

the following example shows that such an extension of mod-retractable rings is not

necessarily mod-retractable.

Example 2.9. Let F be a �eld and put S =

�F F0 F

�. Then matrices E1 =�

1 00 1

�, E2 =

�1 00 0

�and E2 =

�0 10 0

�commutes with the subring R = FE1 �=

F of S and they produce a free base of S a right and left R-module. However every

R-module M is retractable, hence every S module M S is retractable as well by

Theorem 2.7, the S-module E2S is not retractable.

3. Mod-retractable rings

The general case and max rings

First, we prove a general module-theoretic criterion for Mod-retractable rings.

Proposition 3.1. R is a right mod-retractable ring i� for every nonzero module

M and every m 2 M such that mR E M there exists a nonzero homomorphism

M ! mR.

Proof. It su�ces to show the reverse implication. Fix an arbitrary nonzero module

M and its nonzero submodule N . Let n 2 N n f0g. Then an identity mapping

on nR may be extended to the homomorphism � : M ! E(nR). Note that nR E

�(M) � E(nR), hence by the hypothesis there exists a nonzero homomorphism

�(M)! nR � N , which �nishes the proof. �

We recall that

� a ring is called right max provided every nonzero right module contains a

maximal submodule,

� an ideal I � R is right T-nilpotent, provided for every sequence a1; a2; � � � 2 I

there exist n such that anan�1 : : : a1 = 0.

Lemma 3.2. If R is a right max ring, then J(R) is right T-nilpotent.

Proof. It is well known (see for example [1, Remark 28.5] or [25, Proposition 1.8]).

�

Theorem 3.3. If R is a right mod-retractable ring, then R is right max.


Proof. Assume that 0 6= M contains no maximal submodule, �x 0 6= m 2 M and

an arbitrary maximal submodule N of mR. Then M=N contains no maximal sub-

module and so there exists no nonzero homomorphism M=N into a simple mR=N ,

i.e. M=N is not retractable. �

As an immediate consequence of the previous results we obtain

Corollary 3.4. Jacobson radical of every right mod-retractable ring is right T-

nilpotent.

Recall that a torsion theory � = (T ;F) is a pair of classes of modules closed

under isomorphic images such that T \ F = 0, T is closed under taking factors, F

is closed under submodules and for every moduleM there exists a submodule �(M)

for which �(M) 2 T and M=�(M) 2 F . Moreover, a torsion theory is hereditary if

T is closed under submodules.

For a class of right R-modules C, we consider the following annihilator classes:

�C = fM 2 Mod-R j HomR(M; C) = 0g

and

C� = fM 2 Mod-R j HomR(C;M) = 0g:

We notice that the annihilator classes of the form �C for some C � Mod-R coincide

with the torsion classes of modules, and the annihilator classes of the form C�

coincide with the torsionfree classes of modules.

Theorem 3.5. A ring R is mod-retractable if and only if every torsion theory on

Mod-R is hereditary.

Proof. Suppose that R is mod-retractable and � = (T ;F) is a torsion theory. For

M 2 T and N �M , let �(N) be the torsion part of N . Then M=�(N) 2 T , while

N=�(N) 2 F . Then Hom(M=�(N); N=�(N)) = 0. Since N=�(N) is a submodule

of M=�(N) and M=�(N) is retractable, it follows that N=�(N) = 0. Hence N 2 T .

Conversely, suppose that M is an R-module and 0 6= N �M . If Hom(M;N) =

0, then N =2 �(M�). This implies that the torsion theory (�(M�);M�) is not

hereditary. �

A chain (Y�j� � �) is called a strictly decreasing continuous chain of submodules

of Y provided that Y0 = Y , Y� � Y�+1 for each � < �, Y� =S�<� for each limit

ordinal � � �, and Y� = 0.

The following result was proved in [3, Lemma 3].


Lemma 3.6. Let R be a ring and let X and Y be nonzero R-modules. Then the

following are equivalent:

(1) �X � �Y ;

(2) there exists a strictly decreasing continuous chain (Y�j� � �) of submodules

of Y and R-homomorphisms '� : Y� ! X, � < �, such that Y�+1 =

Ker('�) for all � < �.

Theorem 3.7. The following are equivalent for a ring R:

(1) R is mod-retractable

(2) If X E Y then �X = �Y

(3) For every module X, �X = �E(X)

Proof. (1))(2) The inclusion �Y � �X is obvious. In order to prove the converse

inclusion, we will apply Lemma 3.6. So we construct a strictly decreasing continuous

chain (Y�j� � �) of submodules of Y and a family of nonzero homomorphisms

f� : Y� ! X for all � < � .

We put Y0 = Y . Since R is mod-retractable, there is a nonzero homomorphism

f0 : Y0 ! X. Suppose that the submodules Y� and the nonzero homomorphisms

f� : Y� ! X are constructed for all � < �. If � = � + 1 we denote Y� = Ker(f�),

and for � a limit ordinal we put Y� =T�<� Y�. If Y� = 0 then the construction

is �nished. If Y� 6= 0 then Y� \ X 6= 0, hence there is a nonzero homomorphism

f� : Y� ! X such that f�(Y�) � Y� \X.

Since for cardinality reasons there is � with Y� = 0, we can apply Lemma 3.6,

and the proof is complete.

(2))(3) This is obvious.

(3))(1) Let � = (T ;F) be a torsion theory. If X 2 F then T � �X = �E(X),

hence E(X) 2 F . Then F is closed with respect injective envelopes, hence � is

hereditary. �

Commutative rings

To obtain a relation between mod-retractable rings and semiartinian rings in the

commutative case, we state the main theorem of Ohtake in [17, Theorem 8].

Theorem 3.8. Let R be a commutative ring. Then the following are equivalent.

(1) Every torsion theory in Mod-R is of simple type.

(2) Every torsion theory in Mod-R is hereditary.

(3) R is a semiartinian, max ring.

Now, we are able to prove a criterion of mod-retractability for commutative rings.

Note that the the straightforward proof of the converse implication is presented in

[9, Theorem 2.8].


Theorem 3.9. Let R be a commutative ring. Then R is mod-retractable if and

only if R is semiartinian.

Proof. Suppose that R is commutative semiartinian. Then J(R) is T-nilpotent by

[16, Proposition 3.2] and R=J(R) is von Neumann regular by [16, Theoreme 3.1].

Hence R is mod-retractable by [7, Theorem 3] and Theorem 3.8.

The converse is clear from Theorems 3.5 and 3.8. �

Example 3.10. Let � be an ordinal and F a �eld. Put � = max(card �; !).

Examples of commutative semiartinian regular F -subalgebras of the algebra F� of

the socle length � + 1, which are mod-retractable by Theorem 3.9, is constructed

in [5, Theorem 2.6].

Perfect rings

Let M be an R-module. Recall that a submodule N of M is said to be a

super uous in M , denoted by N � M , whenever L � M and M = N + L then

M = L.

Lemma 3.11. Let M be a nonzero semiartinian module, N its super uous sub-

module, and Si, i 2 I, simple modules. If M=N �=L

i2I Si and there exists a

simple subfactor of N which is not isomorphic to any Si, i 2 I, then there exists a

non-retractable factor of M .

Proof. Let T be a simple submodule of N=X where X is a submodule of N that is

not isomorphic to any Si. Since N=X �M=X we may suppose that X = 0.

Assume that there exists a nonzero homomorphism M ! T . Then there exists

a maximal submodule Y � M such that M=Y �= T . If N 6� Y , we get N 6=

N + Y = M and Y = M because N � M , a contradiction. Thus N � Y , which

implies that T �=M=Y is a direct summand ofM=N �=L

i2I Si. Hence there exists

j 2 I such that Si �= T , which contradicts to the hypothesis. We have proved that

Hom(M;T ) = 0. �

Proposition 3.12. Every local ring which is both right and left perfect is right

mod-retractable.

Proof. Assume that R is a local right and left perfect ring. Let M be a right

module over R and N a nonzero submodule of M . Applying [1, Theorem 28.4],

we get that M=MJ(R) �= R=J(R)(�) is nonzero semisimple since R is right perfect

and N is a nonzero semiartinian module because R is left perfect. Thus there

exists a surjective homomorphism M ! M=MJ(R) ! R=J(R) and Soc(N) is a

nonzero submodule of N isomorphic to a direct power of R=J(R), which proved

the existence of a nonzero homomorphism M ! N . �


The following example shows that the assumption "right perfect and left perfect"

in Proposition 3.12 is not super uous, i.e., there exits local, right perfect, but not

left perfect rings, which is neither right nor left mod-retractable.

Example 3.13. Let F be a �eld and VF be an in�nite dimensional vector space

with a countable ordered basis fvn j n 2 Ng, so that every endomorphism of Vk can

be described by a column �nite N � N matrix with entries in k. We denote by I

he identity matrix and by en;m the unit matrices for every n;m 2 N. Consider the

F -subalgebra R of End(VF ) generated by I and all the matrices en;m with n;m 2 N

and n < m. Note that R = fa0I +Pk

i=1 aieni;mij 2 N; a0; : : : ; ak 2 Fg, i.e. R is

the ring of all the N�N upper triangular matrices over the �eld F that are constant

on the diagonal and have only �nitely many nonzero entries o� the diagonal, all

of them over the diagonal. If f is a F -linear combination of �nitely many en;m

with n < m is strictly upper triangular matrix with �nitely many entries, hence

f nilpotent. We can see that all matrices with zero on the diagonal form an ideal

M of R, and every r 2 R nM is invertible. Thus R is a local ring with maximal

ideal M . For every n > 0, we have that e0;1e1;2e2;3:::en�1;n = e1;n 6= 0, so that

M is not left T-nilpotent. It remains to show that M is right T-nilpotent. Take

any sequence a1; a2; : : : in M . Write a1, as a linear combination of the en;m with

n < m: a1 =Pt

i=1 �ieni;mi. We can suppose m1 � m2 � � � � � mt. It is now easy

to verify that amt+1amt: : : a2a1 = 0. Hence M is right T-nilpotent. Note that R

contains no simple right ideal, however it is left semiartinian.

Finally, we will show that R is neither right nor left mod-retractable. First

recall that J(R) is not left T-nilpotent, hence R is not left mod-retractable by

Corollary 3.4.

Suppose that I is an essential right ideal of J(R) and �x i 2 N. Then there exists

m 2 M such that eii+1m =P

j ajeiij 2 I for some pairwise distinct ij and some

aj 6= 0. It implies eii1+s =P

j ajeiija�11 ei1s = eii+1m(a�11 ei1s) 2 I for each s > i1,

hence eii+1Ms 2 I. Since R=Ms is a semiartinian module, we have proved that R=I

is semiartinian as well. Now, let ' : E(RR)! J(R) � E(RR) be a homomorphism.

Since J(R) contains no idempotent, it contains no nonzero injective submodule (cf.

Lemma 3.16), hence kernel of ' is essential in E(R). Thus for every x 2 E(R) there

exists an essential right ideal I of R such that '(xR) �= R=I, which implies that

'(xR) is semiartinian submodule of J(R). Since R contains no simple right ideal,

' = 0, hence Hom(E(R); J) = 0. We have proved that E(R) is not retractable and

so R is not right mod-retractable.

Recall that the ringQ

i2I Ri is right mod-retractable if and only if Ri is right

mod-retractable for each i 2 I, where I is �nite, by [4, Theorem 8] . For perfect

rings we prove more precise structural result.


Theorem 3.14. Let R be a right and left perfect ring. Then the following condi-

tions are equivalent:

(1) R is right mod-retractable;

(2) R is right �nitely mod-retractable;

(3) R �=Q

i�kMni(Ri) for a system of a local right and left perfect rings Ri,

i � k.

Proof. (1))(2) is clear.

(2))(3) We may suppose without lost of generality that R is an indecomposable

ring. Denote by fei; i � ng a complete set of orthogonal idempotents of R. Assume

that there exists i; j � n such that eiR=eiJ(R) 6�= ejR=ejJ(R) and put I = fs �

nj esR=esJ(R) �= eiR=eiJ(R)g. De�ne an idempotent e =P

j2I ei and note that

HomR(eR=eJ; (1� e)R=(1� e)J) = 0 = HomR((1� e)R=(1� e)J; eR=eJ):

Hence either HomR(eR; (1 � e)R) 6= 0 and so (1 � e)R contains a subfactor iso-

morphic to eiR=eiJ or HomR((1 � e)R; eR) 6= 0 and so eR contains a subfactor

isomorphic to ejR=ejJ for a suitable j 62 I, since R is indecomposable. Now ap-

plying Lemma 3.11, either for M = (1� e)R in the �rst case or for M = eR in the

second case, we see that R is not �nitely mod-retractable. Hence eiR=eiJ(R) �=

ejR=ejJ(R) for all i; j. Now it is well known that R �= EndR(e1Rn) �=Mn(e1Re1)

where e1Re1 is a local right and left perfect ring.

(3))(1) follows by [4, Corollary 3 and Theorem 8] and Proposition 3.12. �

Now, we can formulate the following easy structural consequence of Theorems

3.3 and 3.14.

Corollary 3.15. Let R be a left perfect ring. Then R is right mod-retractable if

and only if R �=Q

i�kMni(Ri) for a system of a local rings Ri, i � k, which are

both left and right perfect.

Nonsingular rings

Recall that, a module MR is said to be singular (respectively, nonsingular) if

Z(MR) = MR (respectively, Z(MR) = 0), where Z(MR) = fm 2 M : annrR(m) E

Rg. If Z(RR) = 0 then R is called a right nonsingular ring.

Lemma 3.16. Let R be a right mod-retractable ring and M a non-singular module.

Then

(1) for every nonzero m 2M there exists a nonzero injective module E � mR,

(2) J(M) = 0.


Proof. (1) Fix a nonzero m 2 M . Since E(M) is a retractable module and mR �

E(M), there exists a nonzero homomorphism ' : E(M) ! mR. As M is non-

singular, Ker' is not essential in E(M), hence applying the same technique as in

[2, Lemma 3.3] we can �nd y 2 E(M) such that x = '(y) 6= 0, yR \Ker' = 0 and

so yR �= '(yR). This implies that E(yR)\Ker' = 0 where E(yR) can be expressed

as a direct summand of E(M). As E(yR) �= '(E(yR)), the module '(E(yR)) is

injective.

(2) Letm 2M nf0g. By (1) there exists a nonzero injective submodule E ofmR,

hence there is N �M such thatM = E�N , which implies that J(M) � J(E)�N .

Since R is right max by Theorem 3.3, J(E) 6= E, thus E � mR 6� J(E) �N . We

have proved that m =2 J(M) for arbitrary nonzero m, hence J(M) = 0. �

Corollary 3.17. If R is a right non-singular right mod-retractable ring, then

J(R) = 0 and for every nonzero r 2 R, there exists a nonzero idempotent e 2 rR

such that eR is an injective module.

Theorem 3.18. Every right noetherian right mod-retractable right non-singular

ring is semisimple.

Proof. Let R be a right mod-retractable right non-singular ring. First note that

J(R) = 0 by Corollary 3.17. Assume that R is not semisimple and de�ne two

sequences of right ideals fIng and fJng such that In+1 � Jn+1 = Jn, In 6= 0 and

Jn is not semisimple for each n � 0.

Take a non-trivial idempotent e 2 R which exists by Lemma 3.17. Then either

eR or (1 � e)R is not semisimple. If eR is not semisimple put I0 = (1 � e)R and

J0 = eR otherwise I0 = eR and J0 = (1� e)R.

Since Jn is not semisimple, by Corollary 3.17, there exist an idempotent f 2 Jnsuch that 0 6= fR 6= Jn and fR is injective. Thus Jn = fR � G for a suitable

submodule G and we put Jn+1 = fR and In+1 = G if fR is not semisimple and

Jn+1 = G and In+1 = fR otherwise.

Now we can see thatL

n<! In is an in�nitely generated right ideal. We have

proved that a right mod-retractable right non-singular ring which is not semisimple

is not right noetherian. �

Proposition 3.19. Let R be a semiartinian mod-retractable ring and I be an ideal.

Then

(1) J(R=I) is T-nilpotent,

(2) (R=I)=J(R=I) is non-singular,

(3) every nonzero ideal of R=I contains a nonzero idempotent e 2 R=I such

that e(R=I)=eJ(R=I) is an injective (R=I)=J(R=I)-module.


Proof. (1) It follows by Corollary 3.4

(2) Without lost of generality, we may assume that J(R) = 0 and xSoc(R) = 0

for a nonzero x 2 R. Note that xR \ Soc(R) 6= 0 and take 0 6= y 2 xR \ Soc(R).

Then yRyR = 0, hence yR 2 J(R), a contradiction.

(3) Since R=J(R) is nonsingular by [15, Lemma 7.8], we may apply Corol-

lary 3.17. Thus there exists an idempotent in R=J(R) which can be lifted to an

idempotent e 2 R such that eR=eJ(R) is injective R=J(R)-module. �

Noetherian rings

The following lemma is analogue to [2, Proposition 3.16].

Lemma 3.20. Let R be a right noetherian ring. If R is right mod-retractable, then

it is right artinian and left perfect.

Proof. Since R=N(R) contains no nilpotent ideal, R=N(R) is right non-singular by

[24, Lemma II.2.5]. Note that J(R) � N(R) in general and R=N(R) is semisimple

by Theorem 3.18, which implies that J(R) = N(R). Finally, since J(R) is nilpotent

by [24, Lemma XV.1.4] we get that R is right artinian by Hopkins-Levitzki Theorem

and R is left perfect by [1, Theorem 28.4]. �

Theorem 3.21. Let R be a right noetherian ring. Then R is right mod-retractable

if and only if R �=Q

i�kMni(Ri) for a system of local right artinian rings Ri, i � k.

Proof. ()) It follows from Lemma 3.20 and Corollary 3.15.

(() Since R �=Q

i�kMni(Ri) is left and right perfect, the proof is clear from

Corollary 3.15 �

Acknowledgment. We wish to thank Simion Breaz who formulated and proved

Theorems 3.5 and 3.7 and improved the formulation of Lemma 3.16.

References

[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, 2nd edition, New York1992, Springer.

[2] S. Breaz and J. �Zemli�cka: When every self-small module is �nitely generated, J. Algebra 315(2007), 885{893.

[3] S. Breaz and J. Trlifaj: Modules determined by their annihilator classes, J. London Math.Soc. 81 (2010), 225{240.

[4] S�. Ecevit and M. T. Ko�san: On rings all of whose modules are retractable, Arch. Math.(Brno) 45 (2009), 71{74.

[5] P. C. Eklof, K. R. Goodearl and J. Trlifaj: Dually slender modules and steady rings, ForumMath. 9 (1997), 61{74.

[6] K. R. Goodearl: Von Neumann Regular Rings, London 1979, Pitman, Second Ed. Melbourne,FL 1991, Krieger.

[7] R. M. Hamsher: Commutative rings over which every module has a maximal submodule ,PAMS, 18(6),(1967), 1133{1136.


[8] A. Haghany, M. R. Vedadi, Study of semi-projective retractable modules, Algebra Colloq. 14(2007), no. 3, 489{496.

[9] A. Haghany, O. A. S. Karamzadeh, M. R. Vedadi, Rings with all �nitely generated modulesretractable, Bull. Iranian Math. Soc. 35 (2009), no. 2, 37{45.

[10] M. T. Ko�san, T. K. Lee and Y. Zhou: On modules over group rings, Proc. of the EdinburghMath. Soc., (2010), in press.

[11] S. M. Khuri: Endomorphism rings and lattice isomorphisms, J. Algebra, 56(1979), 401-408.[12] S. M. Khuri: Endomorphism rings of nonsingular modules, Ann. Sci. Math. Qu., 4(1980),

145-152.[13] S. M. Khuri: Nonsingular retractable modules and their endomorphism rings, East-West J.

Math. 2(2000), 161-170.[14] S. M. Khuri: The endomorphism ring of Nonsingular retractable modules, Bull. Aust. Math.

Soc. 43(1)(1991), 63-71[15] T. Y. Lam: Lectures on Modules and Rings, Springer-Verlag, New York, 1991.[16] C. N�ast�asescu and N. Popescu: Anneaux semi-artiniens, Bull. Soc. Math. France, 96 (1968),

357{368.[17] K. Ohtake: Commutative rings over which all torsion theories are hereditary, Comm. Algebra,

9(15) (1981), 1533-4125.[18] P. R�u�zi�cka , J. Trlifaj and J. �Zemli�cka: Criteria of steadiness, Abelian Groups, Module

Theory, and Topology, New York 1998, Marcel Dekker, 359{372.[19] S. T. Rizvi and C. S. Roman: Baer and quasi-Baer modules, Comm. Algebra, 32(1)(2004),

103-123.[20] S. T. Rizvi and C. S. Roman: On K-nonsingular modules and applications, Comm. Algebra,

35(2007), 2960-2982.[21] S. T. Rizvi and C. S. Roman: On direct sums of Baer modules, J. Algebra, 321(2009),

682-696.[22] P. F. Smith, Compressible and related modules, Abelian groups, rings, modules, and homo-

logical algebra, 295313, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, BocaRaton, FL, 2006.

[23] P. F. Smith, M. R. Vedadi, Submodules of direct sums of compressible modules, Comm.Algebra 36 (2008), no. 8, 30423049.

[24] B. Stenstr�om, Rings of Quotients, Die Grundlehren der Mathematischen Wissenschaften,Band 217, Springer-Verlag, New York-Heidelberg, 1975.

[25] A. A. Tuganbaev: Rings whose nonzero modules have maximal submodules, J. Math. Sci.109, 1589-1640.

[26] S. M. Woods, On prefect group rings, Proc. Amer. Math. Soc. 27 (1971), 49-52.[27] J. M. Zelmanowitz, Correspondences of closed submodules, Proc. Am. Math. Soc. 124 (1996),

No.10, 2955-2960.

Department of Mathematics, Gebze Institute of Technology,, C�ayirova Campus 41400

Gebze- Kocaeli, Turkey





I. ON MODULES AND RINGS WITH THE RESTRICTED

MINIMUM CONDITION

M. TAMER KOS�AN AND JAN �ZEMLI�CKA

Abstract. A module M satis�es the restricted minimum condition if M=Nis Artinian for every essential submodule N of M and R is called a right RM-ring whenever RR satis�es the restricted minimum condition as a right module.Several structural necessary conditions for particular classes of RM-rings arepresented in the paper. Furthermore, a commutative ring R is proved to be anRM-ring if and only if R= Soc(R) is Noetherian and every singular module issemiartinian.

1. Introduction

Given a module M over a ring R, recall that N is an essential submodule of M

if there is no non-zero submodule K of M such that K \N = 0 and we say that M

satis�es restricted minimum condition (RMC) if for every essential submodule N of

M , the factor moduleM=N is Artinian. It is easy to see that the class of all modules

satisfying RMC is closed under submodules, factors and �nite direct sums. A ring

R is called a right RM-ring if RR satis�es RMC as a right module. An integral

domain R satisfying the restricted minimum condition is called an RM-domain, i.e.

R=I is Artinian for all non-zero ideals I of R (see [4]). Recall that a Noetherian

domain has Krull dimension 1 if and only if it is an RM-domain [5, Theorem 1].

The purpose of the present paper is to continue in study of works [3],[4], [5],

[10] and [14], in which basic structure theory of RM-rings and RM-domains was

introduced, and the recent paper [1], which describes some properties of classes

of torsion modules over RM-domains known and widely studied for corresponding

classes of abelian groups. As the method of [1] appears to be fruitful, this paper

is focused on study of structure of modules satisfying RMC, in particular singular

ones. For a module M with the essential socle, we show that M satis�es RMC if

and only if M= Soc(M) is Artinian. It is also proved, among other results, that if

M is singular, then M is semiartinian for a module M over a right RM-ring R.

This tools allow us to obtain ring theoretical results for non-commutative as well

as commutative rings. Namely, if R is a right RM-ring and Soc(R) = 0, we prove

that R is a nonsingular ring of �nite Goldie dimension. As a consequence, in Section

2, we obtain characterizations for various classes of right RM-rings via some well

2010 Mathematics Subject Classi�cation. Primary 16D40; Secondary 16E50.Key words and phrases. restricted minimum condition, singular module, semiartinian ring,

semilocal ring, max ring, perfect ring .

132

I. ON MODULES AND RINGS WITH RMC 133

known and important rings (semiartinian, (Von Neumann) regular, semilocal, max,

perfect) plus some (socle �niteness) conditions:

In the case when R is a semilocal right RM-ring and Soc(R) = 0, we show that

R is Noetherian if and only if J(R) is �nitely generated if and only if the socle

length of E(R=J(R)) is at most !. If R is a right max right RM-ring, we prove that

R=Soc(R) is right Noetherian. Section 3 is focused on commutative rings R, it is

shown that such a ring R satis�es RMC if and only if R=Soc(R) is Noetherian and

every singular module is semiartinian.

Throughout this paper, rings are associative with unity and modules are unital

right R-modules, where R denotes such a ring and M denotes such a module. We

write J(R); J(M); Soc(R); Soc(M) for the Jacobson radical of the ring R, for the

radical of the module M , the socle of R and the socle of M , respectively. We also

write N E M and E(M) to indicate that N is an essential submodule of M and

the injective hull of M , respectively.

2. The structure of general right RM-rings

First state one elementary observation about submodules of modules satisfying

RMC and then recall a useful folklore observation (see [11, Lemma 3.6]):

Lemma 2.1. Let K and N be submodules of M such that K E N . If M satis�es

RMC, then N=K is Artinian.

Proof. If we chose a submodules A for which N \ A = 0 and N � A E M , then

K � A E M . Hence M=(K � A) and (N � A)=(K � A) �= N=K are Artinian

modules. �

Let us recall that Goldie dimension of a module M is de�ned as the supremum

of the set

fn 2 N j9N1 : : : ; Nn submodules of M : �ni=1Ni �Mg:

Lemma 2.2. If a module M satis�es RMC, then Goldie dimension of M= Soc(M)

is �nite.

Proof. Put S0 := Soc(M) and �x a submodule S1 of the module M such that

S0 � S1 and S1=S0 = Soc(M=S0). By Zorn's Lemma, we may choose a maximal

set of elements mi 2 M satisfying the condition S1 \L

i2I miR = 0. It is easy

to see that S1 �L

i2I miR E M . SinceL

i2I miR \ S0 = 0, every module miR

has zero socle. Hence miR is not simple and any maximal submodule of miR is

essential in miR. For every i 2 I, let Ni be a �x maximal submodule in miR. AsLi2I Ni E

Li2I miR, the module L = S0 �

Li2I Ni is essential in M . Applying

RMC, we get that M=L is an Artinian module containing an isomorphic copy

of (S1=S0) �L

i2I miR=Ni which implies that I is �nite and S1=S0 is a �nitely

generated semisimple module. By [12, Proposition 6.5], we obtain that the uniform

dimension of M= Soc(M) is �nite. �

134 M. T. KOS�AN AND J. �ZEMLI�CKA

Following [7, Section 7.2] recall that the zero module is the only module of Krull

dimension �1 and the class M� of modules with Krull dimension � is inductively

de�ned as the class of all modules M satisfying two conditions:

(i) M =2S�<�M� ,

(ii) for every decreasing chain M0 �M1 � : : : of submodules of M there exists n

such that Mi=Mi+1 2S�<�M� .

It is easy to obtain information about Krull dimension of modules satisfying RMC.

Proposition 2.3. If an moduleM satis�es RMC, then Krull dimension ofM= Soc(M)

is at most one.

Proof. Note that a module has zero Krull dimension if it is an Artinian module.

Suppose that N0 � N1 � : : : is a sequence of submodules of M= Soc(M). As

M= Soc(M) has a �nite Goldie dimension by Lemma 2.2, there exists n such that

for each i � n either Ni = 0 or Ni+1 E Ni. Since Ni+1=Ni is Artinian by Lemma

2.1, we obtain that M= Soc(M) has a Krull dimension at most 1. �

Obviously, the class of all right RM-rings is closed under taking factors and

�nite products. To show that it is not the case of taking extensions we recall the

notion of a semiartinian module, i.e. a module M such that every non-zero factor

of M contains a non-zero socle. A ring R is called right semiartinian if RR is

a right semiartinian module. Note that every non-zero right module over a right

semiartinian ring is semiartinian (for basic results cf [9]).

Let M be a semiartinian module. It is well known (see e.g. [8, 13]) that every

semiartinian module contains an increasing (so called socle) chain of submodules

(S� j � � 0) satisfying S0 = 0, S�+1=S� = Soc(M=S�) for each ordinal � and

S� =S�<� S� if � is a limit ordinal. Furthermore recall that the �rst ordinal �

such that S� =M is said to be the socle length of M .

Since every semiartinian ring contains essential socle we obtain an easy observa-

tion:

Lemma 2.4. Let R be a right semiartinian ring. Then R is a right RM-ring if and

only if R=Soc(R) is Artinian.

The following example shows that the class of modules satisfying RMC is not

closed under extensions.

Example 2.5. Let R be a right semiartinian ring of socle length 3 and such that

R=Soc(R) is non-Artinian, hence R is not a right RM-ring by Lemma 2.4. Then

R0 = R=Soc(R) is a right RM-ring by Lemma 2.4 because R0= Soc(R0) is semisim-

ple. Clearly Soc(R) satis�es RMC as well. Hence the short exact sequence

0! Soc(R)! R! R=Soc(R)! 0

shows that the class of all modules satisfying RMC is not closed under extensions.


In particular, using constructions of [6], we can �x a �eld F and take R1 as an

F -subalgebra of F -algebra F! of all countable sequences over F generated by ideal

of ultimately zero sequences F (!) where ! denotes the �rst in�nite ordinal, this F -

subalgebra contains exactly ultimately constant sequences. Then R2 is de�ned as

an F -subalgebra of a natural F -algebra R!1 generated by R(!)1 . It is easy to see that

R2 is a right semiartinian ring of socle length 3 and R2= Soc(R2) is non-Artinian.

Let us recall another well-known observation.

Lemma 2.6. Let M be an Artinian R-module. If J(N) 6= N for every nonzero

submodule N of M , then M is Noetherian.

Proof. Assume that M is not Noetherian. Then it contains a semiartinian submod-

ule of in�nite socle length. As M is Artinian, there is a minimal submodule N of

in�nite socle length. Thus N contains no maximal submodule, i.e. J(N) = N . �

Now we are able to clarify structure of RM-rings, which is similar (and in some

sense dual) to structure of semiartinian rings.

Theorem 2.7. Let R be a right RM-ring, S(R) the greatest right semiartinian

ideal of R and put A := R=Soc(R) and S(A) := S(R)=Soc(R).

(i)Tn<! J(A)

n is nilpotent,

(ii) S(A) \ J(A) is nilpotent,

(iii) S(A)=(S(A) \ J(A)) is Noetherian.

Proof. First note that S(A) is the greatest right semiartinian ideal of S(R)= Soc(R).

(i) Since Krull dimension of A is equal to 0 or 1 by Proposition 2.3, we may

directly apply [7, Theorem 7.26], which proved thatTn J(A)

n is a nilpotent.

(ii) PutK := S(A)\(Tn J(A)

n) and I := S(A)\J(A). Note thatK is a nilpotent

by (i). Since S(A) is Artinian by Lemma 2.1, we obtain that I is Artinian. Moreover,

In � J(A)n, and soTn I

n � K. Since I Artinian, there exists n for which In � K,

which �nishes the proof.

(iii) Note that S(A) and soM = S(A)=(S(A)\J(A)) is Artinian and J(M) = 0.

Hence J(N) = 0 for each submodule N of M . By applying Lemma 2.6 we get the

conclusion. �

Corollary 2.8. If Soc(RR) = 0 and J(R)2 = J(R) for a ring R, then R is not a

right RM-ring.

Recall that the singular submodule Z(M) of a module M is de�ned by

Z(M) = fm 2M : mI = 0 for some essential right ideal I of Rg:

The module M is called singular if M = Z(M), and nonsingular if Z(M) = 0. A

ring R is (Von Neumann) regular if for every x 2 R there exists y 2 R such that

x = xyx. Clearly, every regular ring is non-singular (for more properties cf. [15]).

Before we describe structure of singular modules over RM-rings, let us observe that

the structure of regular RM-rings appears to be very lucid:


Proposition 2.9. The following conditions are equivalent for a regular ring R.

(i) R is a right RM-ring,

(ii) R=Soc(R) is Artinian,

(iii) R is semiartinian of socle length 2.

Proof. (i))(ii) By Lemma 2.2,R=Soc(R) is of �nite Goldie dimension. SinceR=Soc(R)

is a regular ring which cannot contain an in�nite set of orthogonal set idempotents,

we obtain that R=Soc(R) is Artinian.

(ii)) (iii) It is obvious because an Artinian regular ring is semisimple.

(iii))(i) It follows from Lemma 2.4. �

Lemma 2.10. Let R be a right RM-ring. Then Z(M) is semiartinian for each

right R-module M .

Proof. Let m 2 Z(M) and put r(m) = fa 2 Ajma = 0g. Then r(m) is an essential

right ideal of R, hence mR �= R=r(m) is Artinian and so semiartinian. �

Theorem 2.11. Let R be a right RM-ring and M be a right R-module.

(i) If M is singular, then M is semiartinian.

(ii) E(M)=M is semiartinian.

(iii) If M is semiartinian, then E(M) is semiartinian. In particular, E(S) is semi-

artinian for every simple module S.

Proof. Assume that M is singular. By Lemma 2.10, Z(M) = M is semiartinian,

hence (i) holds. Since E(M)=M is a singular module by [12, Examples 7.6(3)] and

the class of semiartinian modules is closed under taking essential extensions, (ii)

and (iii) hold. �

Since for a ring R with no simple submodule we obtain Z(R) = 0 by Lemma 2.10,

we can formulate the following consequence of Lemma 2.2:

Corollary 2.12. If Soc(R) = 0 for a right RM-ring R, then R is a nonsingular

ring of �nite Goldie dimension.

Recall that a ring R is called semilocal if R=J(R) is semisimple Artinian.

Lemma 2.13. If R is a semilocal ring, then J(R) + Soc(R) E R.

Proof. Let J(R) + Soc(R) is not essential in R. Then there exists a nonzero right

ideal I � R such that I \ (J(R) + Soc(R)) = 0. Since Soc(I) = Soc(R)\ I = 0 and

R=J(R) contains an ideal which is isomorphic to I, we obtain that Soc(R=J(R)) 6=

R=J(R). Hence R is not semilocal, a contradiction. �

The following example shows that the converse of Lemma 2.13 is not true.

Example 2.14. Suppose that R is a local commutative domain with the maximal

ideal J . It is easy to see that J! is the Jacobson radical of the ring R! and it is

essential in R!, however R! is not semilocal.


Recall that J(R=J(R)) = f0 + J(R)g for an arbitrary ring R.

Proposition 2.15. Assume that R is a right RM-ring.

(i) If Soc(R) = 0, then J(R) E R if and only if R is semilocal.

(ii) If R is a semilocal ring, then J(R)=Soc(J(R)) is �nitely generated as a two-

sided ideal.

Proof. (i) Since J(R) E RR and RR satis�es right RMC, we obtain that R=J(R) is

an Artinian ring. On the other hand, J(R=J(R)) = f0+J(R)g implies that R=J(R)

is semisimple, and hence R is semilocal. The converse follows from Lemma 2.13.

(ii) We note that there exists a �nitely generated right ideal F � J(R) such

that F + (Soc(R) \ J(R)) E J(R) since J(R)=(Soc(R) \ J(R)) has a �nite Goldie

dimension by Lemma 2.2. Thus RF +Soc(R) is a two-sided ideal which is essential

in R as a right ideal by Lemma 2.13. By the hypothesis, R=(RF +Soc(R)) is a right

Artinan ring. As J(R)+Soc(R)=(RF +Soc(R)) is �nitely generated as a right ideal

and

(J(R) + Soc(R))=(RF + Soc(R)) �= J(R)=(J(R) \ (RF + Soc(R)))= J(R)=(RF + (J(R) \ Soc(R)))= J(R)=(RF + Soc(J(R)));

the ideal J(R)= Soc(J(R)) is �nitely generated. �

Note that every Artinian module is semiartinian and recall that ! denotes the

�rst in�nite ordinal.

Lemma 2.16. Let M be an Artinian R-module. Then the following are equivalent:

(i) The socle length of M is greater then !,

(ii) M contains a cyclic submodule with in�nitely generated Jacobson radical,

(iii) M contains a cyclic submodule which is not noetherian.

Proof. (i))(ii) Let M be an Artinian module of nonlimit in�nite socle length and

�x x 2 M such that xR has the socle length ! + 1. Denote the �-th member of

the socle sequence of xR by S�. Since xR is Artinian, we obtain that J(xR) is the

intersection of �nitely many maximal submodules, which implies that xR=J(xR) is

semisimple. As xR=S! is semisimple as well, we have J(xR) � S!. Hence the socle

length of J(xR) is at most !. Assume that J(xR) is �nitely generated. Then the

socle length of J(xR) is non-limit, hence �nite. This implies that xR has a �nite

socle length, a contradiction, i.e. J(xR) is in�nitely generated.

(ii))(iii) Clear.

(iii))(i) As a cyclic non-Noetherian Artinian module is of in�nite non-limit socle

length it have to be greater than !. �

The next result characterizes semilocal right RM-rings further:

Theorem 2.17. If R is a semilocal right RM-ring and Soc(R) = 0. Then the

following conditions are equivalent:


(i) R is right Noetherian,

(ii) J(R) is �nitely generated as a right ideal,

(iii) the socle length of E(R=J(R)) is at most !.

Proof. (i))(ii) Obvious.

(ii))(iii) First, note that every cyclic submodule of E(R=J(R)) is Artinian by

Theorem 2.11. Suppose that the socle length of E(R=J(R)) is greater than !. Hence

it contains an Artinian submodule of the socle length greater than !. By Lemma

2.16, there exists a cyclic module xR with in�nitely generated Jacobson radical.

Fix right ideals I1 and I2 such that xR �= R=I1, I1 � I2 and I2=I1 = J(R=I1). It is

easy to see that I2 is in�nitely generated and J(R) � I2. Since I2=J(R) is a right

ideal of the semisimple ring R=J(R), we obtain that I2=J(R) is �nitely generated,

and hence J(R) is an in�nitely generated right ideal.

(iii))(i) Let I be a right ideal. We show that I is �nitely generated. By Lemma

2.2, there exists �nitely generated right ideals F and G such that F E I, I \G = 0

and F + G E R. First we note that R=(F + G) is an Artinian module with a

submodule isomorphic to I=F and it is also easy to see that R=(F+G) is isomorphic

to a submodule ofL

i�nE(Si) for some simple modules S1; : : : Sn. Since each E(Si)

is isomorphic to some submodule of E(R=J(R)), we obtain that the socle length ofLi�nE(Si) and so of R=(F +G) is at most !. As R=(F +G) is a cyclic module, it

is an Artinian module of �nite socle length, which implies that R=(F + G) is also

a Noetherian module. Therefore I=F and so I are �nitely generated modules. �

Recall that a ring R is called right max if every non-zero right module has a

maximal proper submodule.

Theorem 2.18. If R is a right max right RM-ring, then R=Soc(R) is right Noe-

therian.

Proof. Let I be a right ideal of R=Soc(R). It is enough to show that I is �nitely

generated. If we apply Lemma 2.2 to I, we see that there exists a �nitely generated

right ideal F such that F E I and I=F is Artinian. Since R is a right max ring,

every nonzero submodule of I=F contains a maximal submodule, and so I=F is

Noetherian. By Lemma 2.6, it is �nitely generated. Thus I is �nitely generated as

well. �

As right perfect rings are right max, we get

Corollary 2.19. If R is a right perfect right RM-ring, then R=Soc(R) is right

Noetherian.

The following example shows that a perfect right RM-ring needs not be a (right)

Noetherian ring.

Example 2.20. Let F be a commutative �eld and V be a vector space over F .

Consider the trivial extension R = F �V . Then R is a local ring, hence it is perfect.


The proper ideals of R are the 0�W , where W is an F -subspace of V . Hence the

only essential ideals of R are R and the maximal ideal 0 � V . Then RR satis�es

right RMC. We note that if V is in�nite dimensional, then R is not Noetherian.

Moreover, since every left perfect ring is right Artinian, we can formulate the

following consequence of Lemma 2.4.

Corollary 2.21. If R is a left perfect right RM-ring, then R=Soc(R) is right Ar-

tinian.

3. Characterization of commutative RM-rings

The aim of this section is to generalize results of the work [1] for the class of all

RM-rings.

First, we recall the terminology that we need. Let P be a maximal ideal of a

domain R. For every R-module M , the symbol M[P ] denotes the sum of all �nite

length submodules U of M such that all composition factors of U are isomorphic

to R=P . A module M is self-small, if the functor Hom(M;�) commutes with all

direct powers ofM . Recall thatM is not self-small if and only if there exists a chain

M1 �M2 � ::: �M of submodules such thatSnMn =M and Hom(M=Mn;M) 6=

0 for each n. Denote by Max(M) a set of all maximal submodules of M .

For readers convenience let us formulate results of [1] in one criterion:

Theorem 3.1. [1, Theorem 6, Lemma 3(2), Theorem 9] The following conditions

are equivalent for a commutative domain R:

(i) R is an RM-domain,

(ii) M = �P2Max(R)M[P ] for all torsion modules M ,

(iii) R is Noetherian and every non-zero (cyclic) torsion R-module has an essential

socle,

(iv) R is Noetherian and every self-small torsion module is �nitely generated.

First, make an easy observation.

Lemma 3.2. Every cyclic Artinian module is Noetherian over each commutative

ring.

The following example shows that the assumption of commutativity in Lemma

3.2 is not super uous.

Example 3.3. Let F be a �eld and I = N[f!g be a countable set (I consists of all

natural numbers plus a further index !). The ring R is the ring of non-commutative

polynomials with coe�cients in F and in the non-commutative indeterminates xi,

i 2 I. The cyclic module will be a vector space V over F of countable dimension,

with basis vi, i 2 I, over the �eld F .

We must say how R acts on V . For every n 2 N, set xnvi = vn if i � n and

i 2 N , xnvi = 0 if i < n and i 2 N , xnv! = vn. Moreover, set x!vi = 0 for every


i 2 N , and x!v! = v!. Thus we obtain a left R-module RV . Now RV is cyclic

generated by v! (because xnv! = vn).

The R-submodules of RV are (each one is contained in the following one):

Rv0 � Rv1 � Rv2 � � � � �[

i2N

Rvi � Rv! = V:

Thus the lattice of R-submodules of RV is isomorphic to N[f!g, that is, is order-

isomorphic to the cardinal !+1. Thus the cyclic R-module RR is Artinian but not

Noetherian.

The following observation generalizes [1, Lemma 3(2)].

Theorem 3.4. Let R be a commutative ring. Then R is an RM-ring if and only if

R=Soc(R) is Noetherian and every singular module is semiartinian.

Proof. (:)) Let R be an RM-ring. If A is the greatest semiartinian ideal in R, then

R=A has zero socle and Soc(R) E A. By Lemma 2.1, A=Soc(R) is Artinian, and

so Noetherian by Lemma 3.2. It remains to show that R=A is Noetherian. Without

loss of generality, we may suppose that Soc(R) = 0. Let I be an ideal of R. We

must show that it is �nitely generated. Repeating the argument of the implication

(iii)) (i) of the proof of Theorem 2.17, we can �nd �nitely generated ideals F and

G such that F E I, I \G = 0 and F +G E R. Hence R=(F +G) is Artinian and

it has a submodule which is isomorphic to I=F . Since R=(F +G) is Noetherian by

Lemma 3.2, we have I=F as well as I are �nitely generated. The rest follows from

Lemma 2.10.

((:) Let R=Soc(R) be Noetherian and every singular module be semiartinian.

Fix an ideal I E R. Then R=I is singular and so semiartinian by Lemma 2.10.

Moreover, R=I is Noetherian and semiartinian, hence it is Artinian which �nishes

the proof. �

In light of Theorem 3.4, we ask the following.

Problem 3.5. Is R=Soc(R) Noetherian for each non-commutative right RM-ring

R?

Lemma 3.6. R be a commutative RM-ring and M a singular module. Then M =

�P2 Max(R)M[P ].

Proof. Assume that M 6= �P2Max(R)M[P ] and �x m 2 M n �P2Max(R)M[P ]. Since

M is singular, mR is Artinian and mR �= R=r(m) �=Qr(m)�I AI where AI are

local commutative Artinian rings with maximal ideal I. Since AI �M[I] and there

are only �nitely many I 2 Max(R), we get a contradiction. �

Theorem 3.7. The following conditions are equivalent for a commutative ring R:

(i) R is an RM-ring,

(ii) M = �P2Max(R)M[P ] for all singular modules M ,


(iii) R=Soc(R) is Noetherian and every self-small singular module is �nitely gen-

erated.

Proof. (i) ) (ii) It follows by Lemma 3.6.

(ii)) (i) The proof is the same as the proof of [1, Theorem 6]. Let I be an essen-

tial ideal of R. Then R=I is a cyclic singular module, hence R=I �= �P2Max(R)A[P ]

where all A[P ] are cyclic and only �nitely many A[P ] are non-zero. Since every cyclic

module A[P ] is a submodule of a sum of �nite-length modules, it is Artinian. Thus

R=I is Artinian and R is an RM-ring.

(i))(iii) By Theorem 3.4 and Lemma 2.16, we have R=Soc(R) is Noetherian and

every singular module is semiartinian of socle length less or equal than !. Let M

be a self-small singular module. Then M = �P2Max(R)M[P ] by Lemma 3.6, hence

M[P ] 6= 0 for only �nitely many [P ]. Since Hom(M[P ];M[Q]) = 0 for all P 6= Q,

we may suppose that M = M[P ] for a single maximal ideal P by [16, Proposition

1.6]. Denote by Mi the i-th member of the socle sequence of M . It is easy to see

that Mi = fm 2 M j mP i = 0g. Assume that socle length of M is in�nite, i.e.

Mi 6= Mi+1 and M =Si<!Mi. Then for each i < !, there exist mi 2 Mi+1 nMi

and pi 2 P i such that 0 6= mipi 2 Soc(M). Then multiplication by pi forms a

nonzero endomorphism on M for which Mi � ker pi, a contradiction with the fact

that M is self-small. We have proved there exists n such that Mn = M and so

M has a natural structure of a self-small module over commutative Artinian ring

R=Pn. Hence M is �nitely generated by [2, Proposition 2.9].

(iii))(i) The argument is similar as in the proof of [1, Theorem 9]. If I is an

essential ideal of R, then Soc(R) � I, hence R=I is Noetherian. Moreover, every self-

small module over R=I is singular as an R-module, and so it is �nitely generated.

Now, the conclusion follows immediately from [2, Proposition 3.17]. �

Remark 3.8. Note that Theorem 3.1 is a direct consequence of Theorems 3.4 and

3.7 since singular modules over commutative domains are exactly torsion modules.

References

[1] Albrecht U., Breaz, S.: A note on self-small modules over RM-domains, J. Algebra Appl.13(1) (2014), 8 pages.

[2] Breaz,S., �Zemli�cka, J.: When every self-small module is �nitely generated, J. Algebra 315(2)(2007), 885-893

[3] Chatters, A.W.: The restricted minimum condition in Noetherian hereditary rings, J. Lond.Math. Soc., II. Ser. 4 (1971), 83-87 .

[4] Chatters, A.W., and Hajarnavis, C.R.: Rings with Chain Conditions; Pitman Advanced Pub-lishing 44; Boston, London, Melbourne (1980).

[5] Cohen, I.S.: Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950),27-42.

[6] Eklof, P.C., Goodearl K.R., Trlifaj, J.: Dually slender modules and steady rings, Forum Math.9 (1997), 61{74.

[7] Facchini, A.: Module Theory: Endomorphism Rings and Direct Sum Decompositions in SomeClasses of Modules, Birkh�auser, Basel (1998).

[8] Fuchs, L.: Torsion preradicals and ascending Loewy series of modules, J. Reine Angew. Math.239/240 (1969), 169{179.

[9] Golan, J.S.: Torsion Theories, Longman - Harlow - Wiley, New York 1986.


[10] Huynh, D. V., Dan, P.: On rings with restricted minimum condition, Arch. Math. 51(4)(1988),313-326 .

[11] Jain, S.K., Srivastava, A. K. , Tuganbaev, A. A.: Cyclic Modules and the Structure of Rings,Oxford University Press (2012)

[12] Lam, T. Y.: Lectures on Modules and Rings, Springer-Verlag, New York, (1991).[13] N�ast�asescu, C., Popescu, N.: Anneaux semi-artiniens, Bull. Soc. Math. France 96 (1968), 357[14] Ornstein, A.J. Rings with restricted minimum condition, Proc. Am. Math. Soc. 19 (1968),

1145-1150 .[15] B. Stenstr�om, Rings of Quotients, Die Grundlehren der Mathematischen Wissenschaften,

Band 217, Springer-Verlag, New York-Heidelberg, 1975.[16] �Zemli�cka, J.: When products of self-small modules are self-small. Commun. Algebra, 36(7)

(2008), 2570{2576.

Department of Mathematics, Gebze Institute of Technology, 41400 Gebze/Kocaeli,Turkey

E-mail address: [email protected] [email protected]

Department of Algebra, Charles University in Prague, Faculty of Mathematics andPhysics, Sokolovsk�a 83, 186 75 Praha 8, Czech Republic


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